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additional prerequisites for derivatives involving chain rule inverse functions basic derivative rules power rule product rule quotient rule chain rule

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > derivatives > inverse trig derivatives

You do not need to know the chain rule for the first part of this page, we discuss the basic derivatives first. Once you have learned the chain rule, you can come back here to work the practice problems.

If you want a full length lecture on inverse trig functions and their derivatives, we recommend the following video. The lecturer is one of our favorites and he is very good at explaining the inverse trig functions themselves and their derivatives using plenty of examples.
Note: We stop the video with about 20mins to go since he starts the next topic, integration. If you have already had integration, go ahead and watch the end of the video. It is good too.

### Prof Leonard - Calculus 2 Lecture 6.5: Calculus of Inverse Trigonometric Functions [1hr-31min-10secs]

video by Prof Leonard

Basic Inverse Trig Derivatives (no chain rule)

You might expect the derivatives for inverse trig functions to be similar to derivatives for trig functions but they are not. They are very different. But, again, they appear in similar pairs.

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Important Things To Notice

1. Strangely enough, the derivatives of inverse trig functions do not contain any trig or inverse trig terms. We include some videos below showing the derivation of these equations explaining why this happens.

2. Each pair of inverse trig derivatives are very closely related, even closer than with trig derivatives. Each pair is the same EXCEPT for a negative sign. So, for example, $$[\arccos(t)]' = -[\arcsin(t)]'$$.

3. In the case of the third pair, $$[\arcsec(t)]'$$ and $$[\arccsc(t)]'$$, the denominators contain an absolute value term, $$\abs{t}$$, which is important. Do not leave off the absolute value signs unless you explicitly state that $$t$$ is always positive. Keeping them is always the safe way to make sure you are correct. (Of course, you need to check with your instructor to see what they require.)

4. Remember that the notation $$\sin^{-1}(t)$$ actually means $$\arcsin(t)$$, not $$1/\sin(t)$$.

Before we go on, let's watch some videos showing how the derivatives above come about. It seems strange that when we take the derivative of a function involving inverse trig functions, there are no longer any trig or inverse trig terms. These next videos show how to get the derivatives. Watching these will help you understand trig and inverse trig in more depth.

### PatrickJMT - The Derivative of Inverse Sine or y = arcsin(x) [7min-5secs]

video by PatrickJMT

video by MIP4U

### PatrickJMT - Deriving the Derivative of Inverse Tangent or y = arctan (x) [6min-16secs]

video by PatrickJMT

video by MIP4U

video by MIP4U

### MIP4U - Proof - The Derivative of f(x)=arccsc(x): d/dx[arccsc(x)] [4min-51secs]

video by MIP4U

Here are some practice problems. Calculate the derivative of these functions.

$$y=(1+x^2)\arctan(x)$$

Problem Statement

Calculate the derivative of $$y=(1+x^2)\arctan(x)$$.

Solution

### 2152 solution video

video by PatrickJMT

Inverse Trig Derivatives Using The Chain Rule

As mentioned in the previous section, except for the simplest problems, you will need to use the chain rule for most inverse trig derivatives. Let's look at an example that parallels some of the examples in the previous section.

Evaluate $$[\arcsin(x^2)]'$$

$$\displaystyle{ [\arcsin(x^2)]' = \frac{2x}{\sqrt{1-x^4}} }$$

Problem Statement - Evaluate $$[\arcsin(x^2)]'$$
Solution - For this one, we let $$u = x^2$$. We chose $$x^2$$ since we want $$u$$ to be equal to everything inside the arcsine function.

 $$\displaystyle{ \frac{d}{dx}[\arcsin(x^2)] }$$ let $$u = x^2$$ $$\displaystyle{ \frac{d}{dx}[\arcsin(u)] }$$ Apply the chain rule. $$\displaystyle{\frac{d}{du}[\arcsin(u)] \frac{d}{dx}[u]}$$ $$\displaystyle{\frac{d}{du}[\arcsin(u)] \frac{d}{dx}[x^2]}$$ $$\displaystyle{\frac{1}{\sqrt{1-u^2}} (2x)}$$ $$\displaystyle{\frac{1}{\sqrt{1-(x^2)^2}} (2x)}$$ $$\displaystyle{\frac{2x}{\sqrt{1-x^4}}}$$

Note -
Notice how, when we reverse substituted $$u = x^2$$, we had $$(x^2)^2 = x^4$$. This may be easily missed if you tried to do it in your head.

$$\displaystyle{ [\arcsin(x^2)]' = \frac{2x}{\sqrt{1-x^4}} }$$

Here are some practice problems. Unless otherwise instructed, calculate the derivative of these functions.

$$y=\sin^{-1}(2x+1)$$

Problem Statement

Calculate the derivative of $$y=\sin^{-1}(2x+1)$$.

Solution

### 2155 solution video

video by Krista King Math

$$f(x) = (\sin^{-1}x)^2$$

Problem Statement

Calculate the derivative of $$f(x) = (\sin^{-1}x)^2$$.

Solution

### 2156 solution video

video by PatrickJMT

$$y=\sqrt{\tan^{-1}x}$$

Problem Statement

Calculate the derivative of $$y=\sqrt{\tan^{-1}x}$$.

Solution

### 2150 solution video

video by PatrickJMT

$$y=\tan^{-1}(\sqrt{x})$$

Problem Statement

Calculate the derivative of $$y=\tan^{-1}(\sqrt{x})$$.

Solution

### 2151 solution video

video by PatrickJMT

$$y=e^{\sec^{-1}(t)}$$

Problem Statement

Calculate the derivative of $$y=e^{\sec^{-1}(t)}$$.

Solution

### 2153 solution video

video by PatrickJMT

$$y=\sin^{-1}(x^2+1)$$

Problem Statement

Calculate the derivative of $$y=\sin^{-1}(x^2+1)$$.

Solution

### 2154 solution video

video by PatrickJMT

$$g(t) = \cos^{-1}\sqrt{2t-1}$$

Problem Statement

Calculate the derivative of $$g(t) = \cos^{-1}\sqrt{2t-1}$$.

Solution

### 2157 solution video

video by PatrickJMT

$$\displaystyle{y=\tan^{-1}(x/a) + \ln\sqrt{\frac{x-a}{x+a}}}$$

Problem Statement

Calculate the derivative of $$\displaystyle{y=\tan^{-1}(x/a) + \ln\sqrt{\frac{x-a}{x+a}}}$$.

Solution

### 2158 solution video

video by PatrickJMT

$$y = \sec^{-1}\sqrt{1+x^2}$$

Problem Statement

Calculate the derivative of $$y = \sec^{-1}\sqrt{1+x^2}$$.

Solution

### 2159 solution video

video by PatrickJMT

$$\displaystyle{ y = \sin^{-1}\left( \frac{\cos x}{1+\sin x} \right) }$$

Problem Statement

Calculate the derivative of $$\displaystyle{ y = \sin^{-1}\left( \frac{\cos x}{1+\sin x} \right) }$$.

Solution

### 2160 solution video

video by PatrickJMT