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Find the equation of the tangent line to f(x) = (x^2 - 2) sin x+2x cos x at x =…

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Wize University Calculus 1 Textbook > Derivatives

Tangent Lines

3 Activities

Wize University Calculus 1 Textbook > Derivatives

Derivatives of Trig Functions

6 Activities

Find the equation of the tangent line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x) = (x^2 - 2) \sin x+2x \cos x at x=πx = \pi.
More Tangent Lines Questions:
Solving for the tangent line
Let f(x)=x2+ax−3f(x) = x^2 + ax - 3 be a function such that at x=2x = 2 , the tangent line is parallel to the line given by y=x+3y = x + 3 . Find what aa is and give the equation of the tangent line.
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Derivatives and Tangent Lines
A certain particle is traveling through space and time. We are able to measure the position of the particle at certain time values. Although we can't measure the particle continuously, we know that the particles position function is differentiable. The table gives values of time for the differentiable function s(t) for 0≤t≤40\le t\le4. s(t) represents the position of the particle at time t.
Tangent Lines
Find the point (a,b)\left(a,b\right) at which the tangent line to the curve y=2x2+3x−7y=2x^2+3x-7 is parallel to the tangent line of the curve y=3x4−5x+1y=3x^4-5x+1 at the point (1,−6)\left(1,-6\right).
Solving for the tangent line
Let f(x)=x2+ax−3f(x) = x^2 + ax - 3 be a function such that at x=2x = 2 , the tangent line is parallel to the line given by y=x+3y = x + 3 . Find what aa is and give the equation of the tangent line.
Solving for the tangent line
Let f(x)=x2+ax−3f(x) = x^2 + ax - 3 be a function such that at x=2x = 2 , the tangent line is parallel to the line given by y=x+3y = x + 3 . Find what aa is and give the equation of the tangent line.
Tangent Lines
If f(4)=1f(4) = 1 and f′(4)=−1f'(4) = -1 and g(x)=f(x)x−3g(x) = \dfrac{f(x)}{x-3}, find the equation of the tangent line to the curve g(x)g(x) at x=4x=4 .
Slope of a Tangent Line
Practice: Slope of a Tangent Line
Find all values of xx such that the tangent line to the curve y=2x3−x22−x−5y=2x^3-\frac{x^2}{2}-x-5 is horizontal.
Slope of a Tangent Line
Practice: Slope of a Tangent Line
Find all values of xx such that the tangent line to the curve y=2x3−x22−x−5y=2x^3-\frac{x^2}{2}-x-5 is horizontal.
Slope of a Tangent Line
Practice: Slope of a Tangent Line
Find all values of xx such that the tangent line to the curve y=2x3−x22−x−5y=2x^3-\frac{x^2}{2}-x-5 is horizontal.
Tangent Lines
If f(4)=1f(4) = 1 and f′(4)=−1f'(4) = -1 and g(x)=f(x)x−3g(x) = \dfrac{f(x)}{x-3}, find the equation of the tangent line to the curve g(x)g(x) at x=4x=4 .
Tangent Lines
Is there any point on the graph of y=x2+3x+8y = x^2+ 3x + 8 such that the tangent line is parallel to the line with the equation y=7x+104y = 7x + 104 ?
Tangent Lines
The tangent line to the curve y=−x2−3y=-x^2-3 at x=ax=a passes through (0,1)(0, 1). Find aa and the corresponding function value.
Is there any point on the graph y=x2+3x+8y=x^2 + 3x + 8 such that the tangent line is parallel to the line with equation y=5x+102y = 5x + 102?
Practice: Tangent Through Outside Point
Q:\textbf{Q:} There are points on the curve y=2−x4+xy=\dfrac{2-x}{4+x} at which the tangent line passes through the origin. Find all such points.
Tangent Line Intercept
Let f(x)=3x2−4x+9f(x)=3x^2-4x+9 and let L denote the tangent line to the graph of f(x)f(x) at the point on the graph where x=2x=2. Find the point where L intersects the x-axis.
Tangent Lines
Consider the line y=4x+3y = 4x + 3. To which of the following functions is it tangent at x=1.x = 1.
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Consider the function f(x)=1x2f(x) = \frac{1}{x^2} . Let g(a)g(a) be the xx - intercept of the tangent line of f(x)f(x) at x=ax = a . Find g(a)g(a) for all a≠0a \not = 0 .
Given the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1 , find the equation of the tangent line at x=2x = 2 and find all other points that have tangent lines that are parallel to the one at x=2x = 2 .
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Derivatives and Tangent Lines
A certain particle is traveling through space and time. We are able to measure the position of the particle at certain time values. Although we can't measure the particle continuously, we know that the particles position function is differentiable. The table gives values of time for the differentiable function s(t) for 0≤t≤40\le t\le4. s(t) represents the position of the particle at time t.
Finding Tangent and Normal Lines
Find the tangent line and normal line to f(x)=2x2+x−1f(x)=2x^2+x-1 at x=1x=1.
Tangent Lines
Let f(x)=13x3+12x2+5x−4f(x)=\frac{1}{3}x^3+\frac{1}{2}x^2+5x-4, and suppose g(x) is its inverse function. Find the x-coordinates of all points on the graph of g(x) so that the tangent line at those points are perpendicular to the function y=−7x+2.y=-7x+2.
Tangent Lines
Find a horizontal tangent line of the curve f(x)=x3/3−2x2−5x+4f(x)=x^3/3-2x^2-5x+4
Tangent Lines
Find all points on the graph of f(x)=2x2+x−3f(x)=2x^2+x-3 whose tangent line passes through the point (3,0).
In the function C=−4kt+3C=-4kt+3, where CC represents the concentration of a certain substance in a tank of water, tt represents time, and kk is a positive constant, the rate of change of the dependent variable with respect to the independent variable is a constant. The statement is:
Find the equation of tangent line to y=2x+3y=\sqrt{2x+3} at x=3x=3. Express in slope-intercept form.
Is there any point on the graph y=x2+3x+8y=x^2 + 3x + 8 such that the tangent line is parallel to the line with equation y=5x+102y = 5x + 102?
Tangent Lines
Find the horizontal tangent line of the curve f(x)=x23−2x2−5x+4\displaystyle f(x)=\frac{x^2}{3}-2x^2-5x+4
Tangent Lines
Is there any point on the graph of y=x2+3x+8y = x^2+ 3x + 8 such that the tangent line is parallel to the line with the equation y=7x+104y = 7x + 104 ?
Tangent Lines
The tangent line to the curve y=−x2−3y=-x^2-3 at x=ax=a passes through (0,1)(0, 1). Find aa and the corresponding function value.
Tangent Lines
If f(4)=1f(4) = 1 and f′(4)=−1f'(4) = -1 and g(x)=f(x)x−3g(x) = \dfrac{f(x)}{x-3}, find the equation of the tangent line to the curve g(x)g(x) at x=4x=4 .
Practice: Tangent Through Outside Point
Q:\textbf{Q:} There are points on the curve y=2−x4+xy=\dfrac{2-x}{4+x} at which the tangent line passes through the origin. Find all such points.
Tangent Line Intercept
Let f(x)=3x2−4x+9f(x)=3x^2-4x+9 and let L denote the tangent line to the graph of f(x)f(x) at the point on the graph where x=2x=2. Find the point where L intersects the x-axis.
Find the point on the graph y2 = 4x + 5 where the tangent line is parallel to the line y = 2x + 184.
Tangent Lines
At which point on the graph y=2x+5y=\sqrt{2x+5} the tangent line is parallel to the line with the equation y=13x+11?y=\frac{1}{3}x+11 ?
If f(4)=1f(4)=1, f′(4)=−1f'(4)=-1 and g(x)=f(x)x−3g(x)=\frac{f(x)}{x-3} , then find the equation of the tangent line to the curve g(x)g(x) at x=4x=4.
How many distinct horizontal lines are tangent to the equation f(x)=x2(9−x2)f\left(x\right)=x^2\left(9-x^2\right) ?
🌶️ TOUGH! As shown in the figure below, the tangent line to the graph y = f(x) intersects the x-axis at x = b. What is the value of b in terms of a, f(a), and f’(a)?
Find the point where the tangent line is horizontal for the curve f(x)=x23−2x2−5x+4f(x)=\frac{x^2}{3}-2x^2-5x+4.
Find the yy-intercept of the tangent line of f(x)=1x+x  at  x=2f(x)=\frac{1}{\sqrt{x}}+\sqrt{x} \ \ \text{at }\ x=2.
If f(4)=1f(4) = 1 and f′(4)=−1f'(4) = -1and g(x)=f(x)x−3g(x) = \frac{f(x)}{x-3} then find the equation of the tangent line to the curve g(x)g(x) at x=4x=4.
🌶️ TOUGH! Let f(x)=x2f(x)=x^2, given the tangent line of ff at x=ax=a passes through the point (4,15)(4,15) not on the function, find aa.
Practice: Equation of tangent (and normal) line
Find the tangent and normal lines to the graph f(x)=xf\left(x\right)=\sqrt{x} at x = 9. Which point on this graph is the slope of the tangent line parallel to the line y=3x+1? y = 3x + 1?
Find the equation of the tangent line to the given point for the following function.
f(x)=ex2+x, at P(1,e2)f(x)=e^{x^2+\sqrt{x}},\,\text{at}\,P(1,e^2)
Tangent Lines
Find the point (a,b)\left(a,b\right) at which the tangent line to the curve y=2x2+3x−7y=2x^2+3x-7 is parallel to the tangent line of the curve y=3x4−5x+1y=3x^4-5x+1 at the point (1,−6)\left(1,-6\right).
Tangent Lines
If f(4)=1f(4) = 1 and f′(4)=−1f'(4) = -1and g(x)=f(x)x−3g(x) = \frac{f(x)}{x-3} then find the equation of the tangent line to the curve g(x)g(x) at x=4x=4 ?
Is there any point on the graph y=x2+3x+8y=x^2 + 3x + 8 such that the tangent line is parallel to the line with the equation y=5x+102y = 5x + 102?
Slope of a Tangent Line
Practice: Slope of a Tangent Line
Find all values of xx such that the tangent line to the curve y=2x3−x22−x−5y=2x^3-\frac{x^2}{2}-x-5 is horizontal.
Solving for the tangent line
Let f(x)=x2+ax−3f(x) = x^2 + ax - 3 be a function such that at x=2x = 2 , the tangent line is parallel to the line given by y=x+3y = x + 3 . Find what aa is and give the equation of the tangent line.
Tangent Lines
Consider the function f(x)=−2x2+3x−1f(x) = - 2x^2 + 3x - 1 . There is a point (a,b)(a, b) such that the tangent line to ff at (a,b)(a, b) is perpendicular to the line y=x3+5y = \frac{x}{3} + 5 . Find the point (a,b)(a, b) and write down the equation of the tangent line.
Evaluate the slope of the tangent line to the curve at the given point.
x3+y3=4xy,  at  P(2,2)x^3+y^3=4xy ,\,\, \text{at} \,\, P(2,2)
More Derivatives of Trig Functions Questions:
Quotient with Trig
Find the derivative of h(t)=cos⁡tsin⁡t(1+cos⁡t)\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Find the derivative of 𝑓(𝑥) = 𝑒𝑥 tan 𝑥 + 3.
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Derivatives: Trigonometric Functions
Find f′(π)+f′′(π)f'(\pi)+f''(\pi) if 𝑓(𝑥)=𝑥cos⁡𝑥−sin⁡𝑥𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥 .
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Higher Order Trig Derivative
Find y23y^{23} given y=sin⁡x\displaystyle y=\sin x
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Differentiating functions
Find the derivative of the function
f(x)=sin⁡(x)cos⁡(x)x3f(x) = \frac{\sin(x)\cos(x)}{x^3}
You do not need to simplify your answer.
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Derivatives: Trigonometric Functions
Find the equation of the tangent line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x) = (x^2 - 2) \sin x+2x \cos x at x=πx = \pi.
Derivatives: Trigonometric Functions
Find f′(π)+f′′(π)f'(\pi)+f''(\pi) if 𝑓(𝑥)=𝑥cos⁡𝑥−sin⁡𝑥𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥 .
Derivatives: Trigonometric Functions
Find the derivative of the following function: f(x)=sin⁡xcos⁡x+x3f(x)=\sin x\cos x+\sqrt[3]{x}
Derivatives: Trigonometric Functions
Given that 𝑓(𝑥)=𝑥sin⁡𝑥𝑓(𝑥) = 𝑥\sin 𝑥, find f′ (3π2)f'\ \left(\dfrac{3\pi}{2}\right) .
Derivatives: Trigonometric Functions
Find the derivative of g(x)=sin⁡(πx+1)−cos⁡(πx+32)g\left(x\right)=\sin\left(\pi x+1\right)-\cos\left(\dfrac{\pi x+3}{2}\right)
Derivatives: Trigonometric Functions
Find the derivative of  f(x)=3x2cos⁡(2x)\ f\left(x\right)=3x^2\cos\left(2x\right) .
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Second Derivative with Trig
Find the second derivative for the function g(s)=1tan⁡s+2es−cot⁡scos⁡s\displaystyle g(s)=\frac{1}{\tan s} +2e^{s}-\frac{\cot s}{\cos s}
Higher Order Trig Derivative
Find y23y^{23} given y=sin⁡x\displaystyle y=\sin x
Quotient with Trig
Find the derivative of h(t)=cos⁡tsin⁡t(1+cos⁡t)\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
Tangent Line with Product Rule
Find the equation of the tangent line to the graph of 𝑦=sin⁡𝑥+3𝑥2cos⁡𝑥𝑦 = \sin 𝑥 + 3𝑥^2 \cos 𝑥 at the point where x=π2\displaystyle x=\frac{\pi}{2}
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=ex2sec(x)\displaystyle f(x)=e^{x^2\text{sec}(x)}
Find the derivative of the following function.
f(x)=x2+1+cos⁡x\displaystyle f(x)=\sqrt{x^2+1+\cos{x}}
Derivatives: Trigonometric Functions
Find the slope of the tangent line to the following function at x=π/6:x=\pi/6:
g(x)=e−12ln⁡(sin⁡x)g(x)=e^{-\frac{1}{2}\ln(\sin x)}
Derivatives: Trigonometric Functions
Given that g(1)=g′(1)=1g(1)=g'(1)=1, find f′(π/2) if f(x)=g(sin⁡x)x+1f'(\pi/2) \text{ if }f(x)=\frac{\sqrt{g(\sin x)}}{x+1}
Derivatives: Trigonometric Functions
Find
d22dx22(−10cos⁡(x))\frac{d^{22}}{dx^{22}}(-10\cos(x))
Derivatives: Trigonometric Functions
Find the equation of the tangent line and the normal line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x)=(x^2-2)\sin x+2x\cos x at x=πx=\pi
Derivatives: Trigonometric and Exponential Functions
Compute the derivative of
f(x)=tan⁡(e2x)f(x)=\tan(e^{2x})
Derivatives: Trigonometric Functions
Find the equation of the line tangent to the graph of f(x)=x3sin⁡(πx)f(x)=x^3\sin(\pi x) at the point x=2.x=2.
Practice: Complicated Chain-Rule
Find an expression for the derivative of the following function:
f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos \left(e^{2x}\right)}
Practice: Chain Rule
Find the derivative of the following function:
y=esin⁡(x)y=e^{\sin(\sqrt{x})}
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Find the equation of the tangent line of the graph y=excos⁡ xy=\frac{e^x}{\cos\ x}at the point where 𝑥 = 0.
Derivatives: Trigonometric Functions
Find f′(π)+f′′(π)f'(\pi)+f''(\pi) if 𝑓(𝑥)=𝑥cos⁡𝑥−sin⁡𝑥𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥 .
Derivatives: Trigonometric Functions
Find the derivative of the following function: f(x)=sin⁡xcos⁡x+x3f(x)=\sin x\cos x+\sqrt[3]{x}
Derivatives: Trigonometric Functions
Given that 𝑓(𝑥)=𝑥sin⁡𝑥𝑓(𝑥) = 𝑥\sin 𝑥, find f′ (3π2)f'\ \left(\dfrac{3\pi}{2}\right) .
Derivatives: Trigonometric Functions
Find the derivative of g(x)=sin⁡(πx+1)−cos⁡(πx+32)g\left(x\right)=\sin\left(\pi x+1\right)-\cos\left(\dfrac{\pi x+3}{2}\right)
Derivatives: Trigonometric Functions
Find the equation of the tangent line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x) = (x^2 - 2) \sin x+2x \cos x at x=πx = \pi.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Derivatives: Trigonometric Functions
Find the derivative of  f(x)=3x2cos⁡(2x)\ f\left(x\right)=3x^2\cos\left(2x\right) .
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Higher Order Trig Derivative
Find y23y^{23} given y=sin⁡x\displaystyle y=\sin x
Second Derivative with Trig
Find the second derivative for the function g(s)=1tan⁡s+2es−cot⁡scos⁡s\displaystyle g(s)=\frac{1}{\tan s} +2e^{s}-\frac{\cot s}{\cos s}
Quotient with Trig
Find the derivative of h(t)=cos⁡tsin⁡t(1+cos⁡t)\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
Tangent Line with Product Rule
Find the equation of the tangent line to the graph of 𝑦=sin⁡𝑥+3𝑥2cos⁡𝑥𝑦 = \sin 𝑥 + 3𝑥^2 \cos 𝑥 at the point where x=π2\displaystyle x=\frac{\pi}{2}
The Chain Rule" Derivatives of Trigonometric Functions
Calculate the derivative of the function f(x)=tan⁡(x3)f\left(x\right)=\sqrt{\tan\left(x^3\right)}
Derivatives of Trigonometric Functions
Determine the point(s), if any, where the function y=cos⁡2x−sin⁡2xy=\cos^2x-\sin^2x has a horizontal tangent line.
Derivatives: Trigonometric and Logarithmic Functions
Find ddy[(y5+ln⁡y)tan⁡y]\displaystyle \frac{\text{d}}{\text{d}y}\left[\left(y^5+\ln y\right)\tan y\right].
Derivatives: Trigonometric and Exponential Functions
Evaluate ddx(exsin⁡x)\displaystyle \frac{\text{d}}{\text{d}x}\left(e^x\sin x\right).
Find the derivative of the function f(x)=sin⁡xcos⁡x+x3f(x)=\sin x\cos x+\sqrt[3]{x}.
Find the derivative of 𝑓(𝑥) = 𝑒𝑥 tan 𝑥 + 3.
Given the function f(x)=g(x)h(x)−excos⁡xf\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}-e^x\cos x, g(π2)=2g\left(\frac{\pi}{2}\right)=2, g′(π2)=3g'\left(\frac{\pi}{2}\right)=3, h(π2)=1h\left(\frac{\pi}{2}\right)=1, and h′(π2)=2h'\left(\frac{\pi}{2}\right)=2, find f′(π2)f'\left(\frac{\pi}{2}\right).
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Differentiating functions
Find the derivative of the function
f(x)=sin⁡(x)cos⁡(x)x3f(x) = \frac{\sin(x)\cos(x)}{x^3}
You do not need to simplify your answer.
Practice: Chain Rule
Find the derivative of the following function:
y=sin⁡(x)y=\sin(\sqrt{x})
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Derivatives: Trigonometric Functions
Find f′(π/2)f^{\prime}(\pi/2)if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}, where g(1)=g′(1)=1. g(1)=g^{\prime}(1)=1.
Derivatives: Trigonometric Functions
Find the derivative of the following function.
f(x)=x2+1+cos⁡x\displaystyle f(x)=\sqrt{x^2+1+\cos{x}}
Practice: The Chain Rule
Find the derivative of the following function
y=esin⁡xy=e^{\sin\sqrt{x}}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=tan⁡(arccos(e4x))\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))