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Derivatives/Antiderivatives

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Wize University Calculus 1 Textbook > Integrals

Antiderivatives of Trig Functions

4 Activities

Wize University Calculus 1 Textbook > Derivatives

Derivatives of Exponential Functions

4 Activities

What is the derivative of f(x)=excos⁡xf(x)=e^{x}\cos x?
More Antiderivatives of Trig Functions Questions:
Derivatives/Antiderivatives
What is the derivative of f(x)=e2xcos⁡3xf(x)=e^{2x}\cos3x?
Indefinite Integrals with Trig and Inverse Trig
Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Practice: Indefinite Integral
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Indefinite Integrals with Trig and Inverse Trig
Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Antiderivatives: Trigonometric Functions
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
Find the value of the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\displaystyle \int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx .
Evaluate the indefinite integral ∫sec⁡θtan⁡θ−sin⁡(θ5)dθ\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta.
Evaluate the indefinite integral ∫sec⁡θtan⁡θ−sin⁡(θ5)dθ\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta.
Practice: Indefinite Integral (~F2018 Final Q26)
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Indefinite Integrals with Trig and Inverse Trig
Q:\textbf{Q:} Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Antiderivatives: Indefinite with Trig
Find ∫(tan⁡2x+1sec⁡2x+1+xπ+sin⁡2x)dx\displaystyle\int_{ }^{ }\left(\tan^2x+\frac{1}{\sec^2x}+1+x^\pi+\sin^2x\right)dx
Antiderivatives: Product with Trig
Find dfdθ\displaystyle\frac{df}{d\theta} given f(θ)=(sin⁡θ−cos⁡θ)sec⁡θ\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta
Practice: Indefinite Integral
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
Practice: Indefinite Integral
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Antiderivatives: Trigonometric Functions
Find
∫1+sin⁡3xsin⁡2xdx\int \frac{1+\sin^3 x}{\sin^2 x} dx
Find an antiderivative for
f(x)=−cos⁡x+x2f(x)=-\cos{x}+x^2
Antiderivatives: Trigonometric Functions
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
final114
Find the indefinite integral ∫(1x2+sin⁡x)dx\displaystyle\int\bigg(\frac{1}{x^2}+\sin x\bigg)dx
Antiderivatives: Trigonometric Functions
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
final114
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
Find the exact value of
∫0πsin⁡xdx\int_0^{\pi}\sin xdx
Antiderivatives: Product with Trig
Find dfdθ\displaystyle\frac{df}{d\theta} given f(θ)=(sin⁡θ−cos⁡θ)sec⁡θ\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta
Find the indefinite integral ∫(1x2+sin⁡x)dx\displaystyle\int\bigg(\frac{1}{x^2}+\sin x\bigg)dx . Use C for the integration constant.
Antiderivatives: Indefinite with Trig
Find ∫(tan⁡2x+1sec⁡2x+1+xπ+sin⁡2x)dx\displaystyle\int_{ }^{ }\left(\tan^2x+\frac{1}{\sec^2x}+1+x^\pi+\sin^2x\right)dx
Indefinite Integrals with Trig and Inverse Trig
Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Derivatives/Antiderivatives
Compute the second derivative of f(x)=etan⁡xf(x)=e^{\tan x}.
Find the value of the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\displaystyle \int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx .
Evaluate the indefinite integral ∫sec⁡θtan⁡θ−sin⁡(θ5)dθ\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta.
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Antiderivatives: Trigonometric Functions
Evaluate ∫π3π∣cos⁡x∣dx\int_{\frac{\pi}{3}}^{\pi}\left|\cos x\right|dx.
Antiderivatives: Trigonometric and Exponential Functions
If f′′(x)=cos⁡(2x)−x4+exf''\left(x\right)=\cos\left(2x\right)-x^4+e^x, f(0)=0=f′(0)f\left(0\right)=0=f'\left(0\right)
Practice: Indefinite Integral
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
For each of the following functions, find the most general antiderivative.
f(x)=−cos⁡x+x2f(x)=-\cos{x}+x^2
Find the most general antiderivative.
f(x)=−cos⁡x+x2f(x)=-\cos{x}+x^2
More Derivatives of Exponential Functions Questions:
Derivatives/Antiderivatives
What is the derivative of f(x)=e2xcos⁡3xf(x)=e^{2x}\cos3x?
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos⁡−1(x)−2025)′(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
Finding a Derivative
If f(x)=tan⁡−1(3sin⁡x)f(x)=\tan^{-1}\left(3^{\sin x}\right), find f′(π)f'(\pi).
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Finding an Exponential Derivative
Find the derivative of
ecos⁡(x2)e^{\cos(x^2)}
Find the derivative of 𝑓(𝑥) = 𝑒𝑥 tan 𝑥 + 3.
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Exponential functions: Solving for the tangent line
Let
f(x)=x2−3x+1exf(x)=\frac{x^2-3x+1}{e^x}
Find the equation of the tangent line of f(x)f(x) at x=3x = 3 .
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Exponential functions: Solving for the tangent line
Let
f(x)=x2−3x+1exf(x)=\frac{x^2-3x+1}{e^x}
Find the equation of the tangent line of f(x)f(x) at x=3x = 3 .
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
Finding an Exponential Derivative
Find the derivative of
ecos⁡(x2)e^{\cos(x^2)}
Exponential functions: Solving for the tangent line
Let
f(x)=x2−3x+1exf(x)=\frac{x^2-3x+1}{e^x}
Find the equation of the tangent line of f(x)f(x) at x=3x = 3 .
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Exponential Functions: Evaluating a limit using derivatives
Practice: Evaluating a limit using derivatives
Evaluate lim⁡h→02(2+h)−4h\displaystyle\lim_{h\to 0}\frac{2^{\left(2+h\right)}-4}{h} by finding the derivative of a specific function at a certain point.
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)}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Given f(x)=ex+32xf(x)=e^x+3^{2x} , find f′(2).f'(2).
Derivatives: Exponential and Logarithmic Functions
Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
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}
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}
The chain rule: Logarithmic Derivatives
The derivative of log⁡3(e3x)\log_3\left(e^{3x}\right) is
Derivatives: Exponential and Logarithmic Functions
Find the derivative of the following function.
f(x)=(x2+xln⁡x)exf(x)=(x^2+x\ln{x})e^{\sqrt{x}}
Calculate the derivative of the following
f(x)=etan⁡(x2)\displaystyle f(x)=e^{\tan{(x^2)}}
Finding an Exponential Derivative
Find the derivative of
ecos⁡(x2)e^{\cos(x^2)}
Derivatives: Exponential and Trigonometric Functions
Find the derivative of
f(x)=5x6−sec⁡−1(e2x)+cos⁡−1xf(x)=5x^6-\sec^{-1}(e^{2x})+\cos^{-1}x
Derivatives: Exponential Functions
Find d10dx10(−5x2+4x+50x3−x8−3e3x)\frac{d^{10}}{dx^{10}}(-5x^2+4x+50x^3-x^8-3e^{3x})
Derivatives: Exponential Functions
Find the coordinates of a point on the curve f(x)=xexf(x)=xe^x so that the slope of the tangent line to the curve at this point is parallel to line of x=yx=y.
Derivatives of Exponential Functions
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point (1,e2)(1,e^2)
Derivatives: Trigonometric and Exponential Functions
Compute the derivative of
f(x)=tan⁡(e2x)f(x)=\tan(e^{2x})
Practice: Chain Rule
Find the derivative of the following function:
y=esin⁡(x)y=e^{\sin(\sqrt{x})}
Differentiate the following:
y=ex2+ln⁡xx\begin{aligned} &y=\frac{e^{x^2+\ln x}}{x} \end{aligned}
Differentiate the following:
f(x)=e9−6x(x−5)4e−x2f(x)=\frac{e^{9-6x}}{(x-5)^4e^{-x^2}}
Derivatives: Exponential and Logarithmic Functions
Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.
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)
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Given f(x)=ex+32xf(x)=e^x+3^{2x} , find f′(2).f'(2).
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Tangent Line to Exponential
Find the equation of the line tangent to the graph of the function f(x)=ex2+xf(x)=e^{x^2+\sqrt{x}} at the point P(1,e2)P(1,e^2).
Exponential Functions: Horizontal Tangent
At which point is the tangent line to the graph of y=ex+e−2xy=e^x+e^{-2x} horizontal?
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}
Given f(x)=53xf(x)=5^{3x}, find 𝑓′(0).
Derivatives: Exponential Functions, Inverse Trigonometric Functions
Compute the derivative of f(x)=sec⁡(32x)f(x)=\sec(3^{2x}) .
Derivatives/Antiderivatives
Compute the second derivative of f(x)=etan⁡xf(x)=e^{\tan x}.
Derivatives: Exponential Functions
At which point(s) is the tangent line to the graph y=ex+e−2xy = e^x + e^{-2x} horizontal?
Derivatives: Exponential Functions
Determine ddt(txetan⁡x)\frac{\text{d}}{\text{d}t}\left(txe^{\tan x}\right), where xx is a constant.
If f(x)=2x−1+ex−1f(x)=2x-1 +e^{x-1} find f−1(2)f^{-1}(2)
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 f(x)=22e−2x+2f(x)=\displaystyle \frac{2}{2e^{-2x}+2}.
Find the derivative of f(x)=arcsin⁡(ex+1)f(x)=\arcsin(e^{x+1}).
For the function f(x)=e2xcos⁡xf(x)=e^{2x}\cos\sqrt{x} , find f′(4).f'(4).
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)
Given that f(x)=ecos⁡(ln⁡x)f\left(x\right)=e^{\cos\left(\ln x\right)}, find f′(1)f'\left(1\right).
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}.
Exponential Functions: Evaluating a limit using derivatives
Practice: Evaluating a limit using derivatives
Evaluate lim⁡h→02(2+h)−4h\displaystyle\lim_{h\to 0}\frac{2^{\left(2+h\right)}-4}{h} by finding the derivative of a specific function at a certain point.
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Finding a Derivative
Practice: Finding a Derivative
(tan⁡−1(3sin⁡x))′∣x=π=\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
Each question is worth 3 marks.
Derivatives of Exponential Functions
Compute the derivative of
f(x)=csc⁡(x)+sin⁡(x2)e2x+1f(x) = \frac{\csc(x) + \sin(x^2)}{e^{2x} + 1}
You do not need to simplify your answer.
Exponential functions: Solving for the tangent line
Let
f(x)=x2−3x+1exf(x)=\frac{x^2-3x+1}{e^x}
Find the equation of the tangent line of f(x)f(x) at x=3x = 3 .
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)}
Derivatives: Exponential Functions
If f(x)f(x) is differentiable everywhere, find AA and BB.
f(x)={xe(x2+1) if x≥1Ax+B       if  x<1f(x)=\left\{ \begin{array}{ll} \displaystyle xe^{(x^2+1)}\quad\quad\,\text{if}\,x\geq 1 \\ Ax+B\quad\,\,\,\,\,\,\,\text{if}\,\, x<1\\ \end{array} \right.
Derivatives: Exponential Functions
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)
Derivatives: Exponential Functions
Find the derivative of the following function.
y=esin⁡x+sin⁡ex\displaystyle y=e^{\sin{x}}+\sin{e^x}
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)}
Derivatives of Exponential Functions
Find the derivative of f(x)=x3exf(x)=x^3e^x
Derivatives: Exponential Functions
Calculate the derivative of the following functions.
f(x)=etan⁡(x2)\displaystyle f(x)=e^{\tan{(x^2)}}
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}))