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Tangent lines: Implicit Differentiation

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Wize University Calculus 1 Textbook > Derivatives

Tangent Lines

3 Activities

Wize University Calculus 1 Textbook > Derivatives

Implicit Differentiation

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Wize University Calculus 2 Textbook > Review: Derivatives

Implicit Differentiation

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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
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
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.
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 (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 Implicit Differentiation Questions:
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
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
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit Differentiation
Find the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point (1,9).
Implicit Differentiation
Find y'' if x3+y3=4.x^3+y^3=4.
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for y3=2xy^3=2x.
Implicit Differentiation
Find the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point (1,9).
Implicit differentiation
Find the derivative of the following:
yex+y2=x2ye^{x+y^2}=x^2
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
Find the tangent to the curve 4𝑦2 − 𝑒𝑥 + 𝑥 = 4𝑥 + 3 at the point where 𝑥 = 0 and 𝑦 is positive
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit Differentiation
For the implicit equation x4+x3y−xy2=1x^4+x^3y-xy^2=1, where y = f(x), find y'.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit Differentiation
Find all the equations of the tangent lines to the curve y=x3+x2−xy=x^3+x^2-x that are parallel to the x-axis.
Find dydx\frac{dy}{dx} at x=0x=0 if xcos⁡(y2)=ycos⁡xx\cos\left(\frac{y}{2}\right)=y\cos x.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit Differentiation
Given x2+y=y2−4x^2+y=y^2-4, find y′y'.
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for y3=2xy^3=2x.
Find the point(s) where x2−4x+6y+y2=−9 x^2-4x+6y+y^2 = −9 has horizontal and vertical tangent lines.
Find dydx\dfrac{dy}{dx} for the function ycos⁡x=x2+cos⁡yy\cos x=x^2+\cos y
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Practice: Tangent and Normal Lines with Implicit
Q:\textbf{Q:} Find the equation of the tangent and the normal lines to the curve x2cos⁡2y−sin⁡y=0x^2\cos^2y-\sin y=0 at the point (0,π)(0,\pi).
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit with Trig
Find dy/dxdy/dx for xy=tan⁡(x+y2)\displaystyle xy=\tan(x+y^{2})
Implicit Differentiation
Let xx, yy satisfy the equation xy=6y−x+1x^y = 6y - x + 1. Find dydx\frac{dy}{dx} at the point (x,y)=(1,1/6)(x, y) = (1, 1/6).
Derivatives: Logarithmic Functions
Compute the derivative of f(x)=xx+1f(x) = x^{x + 1}. Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
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
Implicit Differentiation: Logarithmic Functions
For the following equations, solve for dydx\frac{dy}{dx} :
a) y=(5x2+3)sin⁡(x)y = (5x^2 + 3)^{\sin(x)}
b) x2y+3y2x2=x+yx^2y + 3y^2 x^2 = x + y
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)
Implicit differentiation
Find the derivative of the following:
yex+y2=x2ye^{x+y^2}=x^2
Implicit differentiation
Find the derivative of the following:
sin⁡(x+y)−sec⁡(x2+y2)=y\sin(x+y)-\mathrm{\sec}(x^2+y^2)=y
Implicit Differentiation
Evaluate the slope of the tangent line to the curve at the given point.
x3+y3=ysin⁡x+1,  at  P(0,1)x^3+y^3=y\sin{x}+1,\,\, \text{at}\,\,P(0,1)
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)
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)
Find the equation of the tangent line to the equation x3+ln⁡y=8x^3+\ln y=8 at the point (2, 1)
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)
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Implicit Differentiation
Find the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point (1,9).
Implicit Differentiation
Find y'' if x3+y3=4.x^3+y^3=4.
Find dydx\frac{dy}{dx} given that exy−xy=2x2+1 e^{xy}-xy=2x^2+1
final114
Find y′y' for the function x4+x3y−xy2=1x^4+x^3y-xy^2=1
Differentiate the following functions.
(a) g(t)=(t5+t2)6tg(t)=(t^5+t^2)^{6t} (b) f(x)=x6xf(x)=\sqrt x^{6x} (c) f(x)=(x3+4)tan⁡xf(x)=(x^3+4)^{\tan x} (d) Finddydx\frac{dy}{dx} if ycos⁡(x)=xtan⁡(y)y\cos(x)=x\tan(y)
Differentiate y with respect to x in the equation 2x2+3xy−y2=12x^2+3xy-y^2=1.
Find dydx\frac{dy}{dx} given that exy−xy=2x2+1 e^{xy}-xy=2x^2+1
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)
Evaluate the slope of the tangent line to the curve at the given point.
x3+y3=ysin⁡x+1,  at  P(0,1)x^3+y^3=y\sin{x}+1,\,\, \text{at}\,\,P(0,1)
Implicit Differentiation
Given x2+y=y2−4x^2+y=y^2-4, find y′y'.
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for y3=2xy^3=2x.
Find the point(s) where x2−4x+6y+y2=−9 x^2-4x+6y+y^2 = −9 has horizontal and vertical tangent lines.
Find dydx\dfrac{dy}{dx} for the function ycos⁡x=x2+cos⁡yy\cos x=x^2+\cos y
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Practice: Tangent and Normal Lines with Implicit
Q:\textbf{Q:} Find the equation of the tangent and the normal lines to the curve x2cos⁡2y−sin⁡y=0x^2\cos^2y-\sin y=0 at the point (0,π)(0,\pi).
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit with Trig
Find dy/dxdy/dx for xy=tan⁡(x+y2)\displaystyle xy=\tan(x+y^{2})
Find the slope of the tangent line to the curve ln⁡(x−y)+3y2=3\ln(x-y)+3y^2=3 at point (3,2)(3,2).
Find y′y' for the function x4+x3y−xy2=1x^4+x^3y-xy^2=1
Implicit Differentiation: Estimate of Implicit Function
Use a linear approximation to estimate the yy value of the curve described by x3+3xy+y3=5x^3+3xy+y^3=5 at the point for which x=0.1x=0.1 (leave it as an exact value if you're not allowed a calculator on the exam).
Find the point on the graph y2 = 4x + 5 where the tangent line is parallel to the line y = 2x + 184.
Concept Clarifier
Compute the second derivative of the implicit function y3=xcos⁡yy^3=x \cos y, where yy is a function of xx. Do not simplify.
Find the point(s) where x2−4x+6y+y2 = −9 has horizontal and vertical tangent lines.
Find the slope of the tangent line to the curve ln(x − y) + y2 = 4 at point (3,2).
(Express your answer as a fraction in lowest terms. If the answer is a negative number, put the negative sign in front of the entire fraction.)
Find the equation of the tangent line at the point (1,2)(1,2) of the curve y3−2x5y2=yln⁡(x)y^3-2x^5y^2=y\ln(x).
Find the derivative of sin⁡(x+y)−sec(x2+y2)=y\sin(x+y)-\mathrm{sec} (x^2+y^2)=y.
Find the derivative of y with respect to x of the following equation, y2+x5y3=5−ln⁡yy^2+x^5y^3=5-\ln y.
Given that x2+y=y2−4x^2+y=y^2-4, where y is dependent on x, find dydx\frac{dy}{dx}.
Implicit Differentiation
For the implicit equation x4+x3y−xy2=1x^4+x^3y-xy^2=1, where y = f(x), find y'.
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)
Find the tangent to the curve 4𝑦2 − 𝑒𝑥 + 𝑥 = 4𝑥 + 3 at the point where 𝑥 = 0 and 𝑦 is positive
Implicit Differentiation
Find all the equations of the tangent lines to the curve y=x3+x2−xy=x^3+x^2-x that are parallel to the x-axis.
Find dydx\frac{dy}{dx} at x=0x=0 if xcos⁡(y2)=ycos⁡xx\cos\left(\frac{y}{2}\right)=y\cos x.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit differentiation
Suppose 3x3+4y2=193x^3+4y^2=19, where x, yx,\ y are both functions of tt.
If dydt=3\frac{dy}{dt}=3, find dxdt\frac{dx}{dt} when y=2y=2.
Solve for dydx\frac{dy}{dx} :
x2y+3y2x2=x+yx^2y+3y^2x^2=x+y
Implicit Differentiation
Use implicit differentiation to calculate the derivative of
θ(x)=arctan⁡(x)\theta(x) = \arctan(x)
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for ln⁡(x2+y2)+2xy=4\ln(x^2+y^2)+2xy=4.
Implicit Differentiation
Evaluate the slope of the tangent line to the curve at the given point.
x3y2+xtan⁡y=4 at P(x,y)\displaystyle x^3y^2+x\tan{y}=4 \, at\, P(x,y)
Evaluate the slope of the tangent line to the curve at the given point.
x2y3+xln⁡y+ysin⁡x=10  at  P(x,y)\displaystyle x^2y^3+x\ln{y}+y\sin{x}=10\,\, at \,\,P(x,y)
Practice: Implicit Differentiation
Find dydx\frac{dy}{dx} for the following
tan⁡yx=x\tan{\frac{y}{x}}=x
More Implicit Differentiation Questions:
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
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
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
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
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
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit Differentiation
For the implicit equation x4+x3y−xy2=1x^4+x^3y-xy^2=1, where y = f(x), find y'.
Implicit Differentiation
Use implicit differentiation to calculate the derivative of
θ(x)=arctan⁡(x)\theta(x) = \arctan(x)