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Linear Approximation and Trigonometric Derivative

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Wize University Calculus 1 Textbook > Derivatives

Derivatives of Inverse Trig Functions

3 Activities

Wize University Calculus 1 Textbook > Applications of Differentiation

Linearization (Linear Approximation)

4 Activities

a) Show that for f(x)=arcsin⁡(x)f(x) = \arcsin(x) , that f′(x)=11−x2f'(x) = \frac{1}{\sqrt{1 - x^2}} .


b) Use linear approximation at a suitable close value to estimate arcsin⁡(0.1)\arcsin(0.1) . Your solution may be left in terms of fractions.

More Derivatives of Inverse Trig Functions Questions:
Inverse trigonometric derivative
Find the derivative of g(x)=(cos⁡−1x)(sin⁡−1x)g(x)=\left(\cos^{-1}x\right)\left(\sin^{-1}x\right).
Derivatives: Inverse Trigonometric Functions
Given that f(x)=2−arcsin⁡(x)f\left(x\right)=2-\arcsin\left(x\right), find f′(0)f'\left(0\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin⁡(x+2)f\left(x\right)=\arcsin\left(x+2\right)
Derivatives: Inverse Trigonometric Functions
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right)
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos⁡−1(x)−2025)′(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
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)'
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Inverse Trigonometric Functions
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
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
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Derivatives: Inverse Trigonometric Functions
Find [tan⁡−1(x3)]′[\tan^{-1}(\sqrt[3]x)]'
Derivatives: Inverse Trigonometric Functions
Show thatddxsin⁡−1x=11−x2\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
Practice: Derivative of Inverse Trig
Q:\textbf{Q:} Show that ddxcos⁡−1x=−11−x2.\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
Practice: Tangent Line with Inverse Trig
Q:\textbf{Q:} Find the equation of the tangent line to the graph of f(x)=cot⁡−1(x2)\displaystyle f(x)=\cot^{-1}(x^{2}) at the point x=2x=2.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
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
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Derivatives: Inverse Trigonometric Functions
Find f′(x)f'(x) if f(x)=arctan⁡xf(x)=\arctan \sqrt{x}
Differentiate the following
y=tan⁡−1x2−1y=\tan^{-1}\sqrt{x^2-1}
Find the derivative of the following function.
f(x)=arcsinx\displaystyle f(x)=\sqrt{\text{arcsin{x}}}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Derivatives: Inverse Trigonometric Functions
Find [tan⁡−1(x3)]′[\tan^{-1}(\sqrt[3]x)]'
Derivatives: Inverse Trigonometric Functions
Show thatddxsin⁡−1x=11−x2\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
Practice: Derivative of Inverse Trig
Q:\textbf{Q:} Show that ddxcos⁡−1x=−11−x2.\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
Practice: Tangent Line with Inverse Trig
Q:\textbf{Q:} Find the equation of the tangent line to the graph of f(x)=cot⁡−1(x2)\displaystyle f(x)=\cot^{-1}(x^{2}) at the point x=2x=2.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Exponential Functions, Inverse Trigonometric Functions
Compute the derivative of f(x)=sec⁡(32x)f(x)=\sec(3^{2x}) .
Derivatives: Logarithmic and Inverse Trigonometric Functions
Evaluate ddx(sin⁡−1xsin⁡xlog⁡3x)\displaystyle \frac{\text{d}}{\text{d}x}\left( \frac{\sin^{-1}x\sin x}{\log_3x}\right).
Find the derivative of f(x)=arcsin⁡(ex+1)f(x)=\arcsin(e^{x+1}).
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Derivatives: Inverse Trigonometric Functions
Find the derivative of f(x)=(arccos⁡x)2sec⁡xf\left(x\right)=\frac{\left(\arccos x\right)^2}{\sec x}.
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\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)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Finding a Derivative
Practice: Finding a Derivative
(tan⁡−1(3sin⁡x))′∣x=π=\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
Derivatives: Inverse Trigonometric Functions
Find the derivative of the following function.
f(x)=1arcsin(x)+arcsin(1x)\displaystyle f(x)=\frac{1}{\text{arcsin}(x)}+\text{arcsin}\left(\frac{1}{x}\right)
Inverse Trigonometric Functions
Calculate the derivative of the following functions.
f(x)=arcsin(ex+1)\displaystyle f(x)=\text{arcsin}(e^{x+1})
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}))
More Linearization (Linear Approximation) Questions:
Use linearization to construct a linear approximation to x4\sqrt[4]x for xx values near 11, given that f′(x)=14x−34f'(x)=\frac14x^{-\frac34}.
Linear Approximation
Using a linear approximation, estimate f(4.1)f(4.1), given that f(4)=2f(4)=2 and f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}. If f(x)f(x) is concave up near this point, what does that tell us about our estimate?
Linear Approximation
Use a suitable linear approximation to estimate 3.99\sqrt{3.99}. Is your estimate likely to be an over- or under-estimate? HINT: Use f(x)=xf(x)=\sqrt{x}, which gives us f′(x)=12xf'(x)=\frac1{2\sqrt{x}} and f′′(x)=−14x23f''(x)=-\dfrac{1}{4\sqrt[3]{x^2}}.
Linear Approximation of Given Function
Use a linear approximation to estimate f(4.1)f(4.1), given that f(4)=2f(4)=2 and f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}.
Linear Approximation
Using a linear approximation, estimate f(4.1)f(4.1), given that f(4)=2f(4)=2, f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}
Linear Approximation
Use a suitable linear approximation to estimate 3.99\sqrt{3.99}.
Practice: Linear Approximation
Consider the function
f(x)=sin⁡(x)f(x) = \sin(x)
a) Find the linear approximation of the function f(x)f(x) near x=0x = 0 .
Written Answer 5
Use linear approximation at a suitable close value to estimate arctan⁡(0.9)\arctan(0.9). Your answer may be left in terms of fractions.
Linear Approximation
Approximate (28)1/3(28)^{1/3} using a linear approximation of the function h(x)=x1/3h(x) = x^{1/3}
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value? Estimate the max error bound for this approximation.
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value?
The linearization of f(x)=tanxf(x)=tanx at x=π4x=\frac{\pi}{4} is
Linear Approximation
Approximate 93.\sqrt[3]{9}.
Linear Approximation
Use a suitable linear approximation to estimate 1.9951.99^5
Linear Approximation
Use a suitable linear approximation to estimate 3.99\sqrt{3.99}.
Linear Approximation
Use a linear approximation to estimate 9.2\sqrt{9.2} using f(x)=3+x.f(x)=\sqrt{3+x}.
Linear Approximation
The linear approximation L(x)L(x) of the function f(x)=xf(x)=\sqrt{x} at x=16x=16 is
Practice: Linear Approximations
Practice: Linear Approximations
Approximate ln⁡(1.01)\ln\left(1.01\right) and ln⁡(0.98)\ln\left(0.98\right).
Find the linear approximation of the following function, and determine if it is an over/under approximation.
f(x)=x34+x3f(x)=\frac{x^3}{4+x^3} at x=2.1x=2.1
final114
The linear approximation L(x)L(x) of the function f(x)=xf(x)=\sqrt{x} at x=16x=16 is
Approximate x32+x23 x^{\frac{3}{2}}+x^{\frac{2}{3}} at x=1.1x=1.1 using a linear approximation.
final114
The linear approximation L(x)L(x) of the function f(x)=xf(x)=\sqrt{x} at x=16x=16 is (enter L=...)
Linear Approximation: Estimate using Given Values
Use a linear approximation to estimate f(2.01)f(2.01) given that f(2)=1f(2)=1 and f′(2)=2f^{\prime}(2)=2.
Linear Approximation
Use a suitable linear approximation to estimate 8.01238.01^{\frac{2}{3}}.
Linear Approximation: Estimating a Cube Root
Use a linear approximation to estimate 653 \sqrt[3]{65}. Then use the same linearization equation to estimate 283\sqrt[3]{28}. Is it still a good approximation? Why?
Linear Approximation of Given Function
Use a linear approximation to estimate f(4.1)f(4.1), given that f(4)=2f(4)=2 and f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}.
Linear Approximation: Estimating a 4th Root
Use a suitable linear approximation to estimate 68.821/468.82^{1/4} (leave it as an exact value if you're not allowed a calculator on the exam).
Use linearization to construct a linear approximation to x4\sqrt[4]x for xx values near 11.
Use a linear approximation to estimate 99.9\sqrt{99.9}.
Concept Clarifier
Use linear approximation to estimate the value of 603\sqrt[3]{60}.
Linear Approximation
The linear approximation L(x)L(x) of the function f(x)=cos⁡xf(x)=\cos{x} at x=π3x=\frac{\pi}{3} is
Linear Approximation
Using a linear approximation, estimate f(4.1)f(4.1), given that f(4)=2f(4)=2 and f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}. If f(x)f(x) is concave up near this point, what does that tell us about our estimate?
Approximate arctan⁡(0.9)\arctan\left(0.9\right) by linear approximation.
Use a linear approximation to approximate the value of the function by using the given information.
f(2)=1,f′(2)=2f(2)=1, f^{\prime}(2)=2 Approximate f(2.01)f(2.01). Enter your answer to two decimal places.
Use a linear approximation to estimate the value of the following function at the given point.
f(x)=2x+1x3+1\displaystyle f(x)=\frac{2x+1}{x^3+1} Approximate f(1.1)f(1.1).
Use a linear approximation to estimate 821/482^{1/4}.
Approximate x32+x23 x^{\frac{3}{2}}+x^{\frac{2}{3}} at x=1.1x=1.1 using linear approximation about x0=1x_0=1.
Use a linear approximation to estimate the given quantity.
8.012/38.01^{2/3}
Practice: Relative Error
A farmer used 100m of fence material to create a square pen to hold in his animals. However, there's a possible error of 2m in the length of the fence he used.
a) Use a linear approximation to estimate the maximum possible absolute error in the area of the pen. Include units in the answer.
b) Use the previous linear approximation to find the maximum relative error in the area. Include units in your answer.
Practice: Linear Approximation
Consider the function
f(x)=sin⁡(x)f(x) = \sin(x)
a) Find the linear approximation of the function f(x)f(x) near x=0x = 0 .
Linear Approximation
Consider the function
f(x)=4−xf(x) = \sqrt{ 4 - x}
Practice: Relative Error
A farmer used 100m of fence material to create a square pen to hold in his animals. However, there's a possible error of 2m in the length of the fence he used.
a) Use a linear approximation to estimate the maximum possible absolute error in the area of the pen. Include units in the answer.
b) Use the previous linear approximation to find the maximum relative error in the area. Include units in your answer.
Practice: Linear Approximation
Consider the function
f(x)=sin⁡(x)f(x) = \sin(x)
a) Find the linear approximation of the function f(x)f(x) near x=0x = 0 .
Consider the function
f(x)=4−xf(x) = \sqrt{ 4 - x}
a) Find the linear approximation of the function f(x)f(x) near x=0x = 0 .
Using a linear approximation, estimate g(1.1)g(1.1), given that g(x)=f(x2+1)g(x)=f(x^2+1) and f(2)=f′(2)=1f(2)=f^{\prime}(2)=1.
Use a linear approximation to estimate
8.012/38.01^{2/3}
For the following exercises, use a linear approximation to estimate the given quantities.
8.012/38.01^{2/3}