Find the derivative of the given function.f(x)=(x3+4x)sin−1xf(x)=(x^3+4x)\sin^{-1}xf(x)=(x3+4x)sin−1x

x3+4x1−x2+(3x2+4)sin⁡−1x\frac{$$x^3+4x$$}{\sqrt{$$1-x^2$$}}+\$$left(3x^2+4$$\right)\$$sin^{-1}x1$$−x2x3+4x+(3x2+4)sin−1x

Find the derivative of the given function. f ( x ) = ( x 3 + 4 x ) sin − 1 x f(x)=(x^3+4x)\sin^{-1}x f ( x ) = ( x 3 + 4 x ) sin − 1 x

Here, we wanna find the derivative of this function f of x. That is x cubed+4x times the inverse sine of x. So in finding the derivative here, f prime of x, we need to use the product rule since I have one function multiplying another function. Remember our memory tool for the product rule, of course, left d right plus a right d left. That's our left hand function. So here, x cubed plus 4x times the derivative of that right hand function. Now we just learned that the derivative of the inverse sine of x is going to be 1 over the square root of 1 minus x squared. Then continuing on with our product rule here, plus right d left, so that's the inverse sine of x, times the derivative of this x cubed plus 4 x, that is 3 x squared plus 4. Now I can write this a little bit nicer because I can go ahead and condense this all into one fraction. That is x cubed plus 4 x over the square root of 1 minus x squared. And then plus here, we typically like to have all of our x terms in the front. So this is 3x squared plus 4 times the inverse sine of x. And this is my full derivative here, f prime of x. So as we learn more and more rules for finding derivatives, we can continue to combine them with all of the rules that we've already learned. Like here, we used our product rule along with our new derivative rule for finding the derivative of inverse sine. If you have any questions, let us know, and I'll see you in the next one.

Find the derivative of the given function.f(x)=(x3+4x)sin⁡−1xf(x)=$$(x^3+4x$$)\$$sin^{-1}xf(x)=(x3+4x)sin$$−1x

x3+4x1−x2+(3x2+4)sin⁡−1x\frac{$$x^3+4x$$}{\sqrt{$$1-x^2$$}}+\$$left(3x^2+4$$\right)\$$sin^{-1}x1$$−x2x3+4x+(3x2+4)sin−1x

3x2+41−x2\frac{$$3x^2+4$$}{\sqrt{$$1-x^2$$}}1−x23x2+4

(3x2+4)sin⁡−1⁡x+(x3+4x)cos⁡−$$1x(3x^2+4$$)\$$sin^{-1}$$⁡$$x+(x^3+4x$$)\$$cos^{-1}x(3x2+4)sin$$−1⁡x+(x3+4x)cos−1x

(3x2+4)sin⁡−1⁡x−(x3+4x)cos⁡−$$1x(3x^2+4$$)\$$sin^{-1}$$⁡$$x-(x^3+4x$$)\$$cos^{-1}x(3x2+4)sin$$−1⁡x−(x3+4x)cos−1x

Table of contents

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0. Functions7h 55m

Introduction to Functions
18m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
36m

Exponential & Logarithmic Equations
35m

Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation3h 18m

Basic Rules of Differentiation
39m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

Derivatives of Trig Functions
40m

The Chain Rule
49m

4. Applications of Derivatives2h 38m

Motion Analysis
36m

Implicit Differentiation
27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
23m

Finding Global Extrema
50m

The First Derivative Test
1h 5m

Concavity
1h 1m

The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
1h 52m

Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
24m

7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

Integrals of Trig Functions
15m

Initial Value Problems
27m

8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

Substitution
1h 19m

9. Graphical Applications of Integrals2h 27m

Area Between Curves
1h 18m

Introduction to Volume & Disk Method
1h 8m

10. Physics Applications of Integrals 3h 16m

Kinematics
1h 13m

Work
2h 3m

11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 31m

Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
52m

12. Techniques of Integration7h 41m

Integration by Parts
2h 32m

Partial Fractions
4h 29m

Improper Integrals
39m

13. Intro to Differential Equations2h 55m

Basics of Differential Equations
48m

Slope Fields
19m

Euler's Method
22m

Separable Differential Equations
1h 25m

14. Sequences & Series5h 36m

Sequences
1h 22m

Review of Factorials
11m

Series
44m

Convergence Tests
3h 17m

15. Power Series2h 19m

Introduction to Power Series
1h 9m

Taylor Series & Taylor Polynomials
1h 10m

16. Parametric Equations & Polar Coordinates7h 58m

Parametric Equations
1h 6m

Calculus with Parametric Curves
1h 13m

Polar Coordinates
2h 5m

Calculus in Polar Coordinates
55m

Conic Sections
2h 36m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Inverse Trigonometric Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Inverse Trigonometric Functions

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Multiple Choice

Find the derivative of the given function.
$f(x)=(x^3+4x)$\sin$^{-1}x$

$$\frac{x^3+4x}{\sqrt{1-x^2}$}+$\left$(3x^2+4$\right$)$\sin$^{-1}x$

$$\frac{3x^2+4}{\sqrt{1-x^2}$}$

$(3x^2+4)$\sin$^{-1}x+(x^3+4x)$\cos$^{-1}x$

$(3x^2+4)$\sin$^{-1}x-(x^3+4x)$\cos$^{-1}x$

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Verified step by step guidance

Identify the function f(x) = (x^3 + 4x) $\sin$^{-1}x. This is a product of two functions: u(x) = x^3 + 4x and v(x) = $\sin$^{-1}x.

Apply the product rule for differentiation, which states that if you have a function f(x) = u(x)v(x), then its derivative f'(x) = u'(x)v(x) + u(x)v'(x).

Differentiate u(x) = x^3 + 4x. The derivative u'(x) is found by applying the power rule: u'(x) = 3x^2 + 4.

Differentiate v(x) = $\sin$^{-1}x. The derivative v'(x) is $\frac{1}{\sqrt{1-x^2}$}, which is the standard derivative of the inverse sine function.

Substitute u'(x), v(x), u(x), and v'(x) into the product rule formula: f'(x) = (3x^2 + 4)$\sin$^{-1}x + (x^3 + 4x)$\frac{1}{\sqrt{1-x^2}$}.

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Struggling with Calculus?

Find the derivative of the given function.f(x)=(x3+4x)sin−1xf(x)=(x^3+4x)\(\sin\)^{-1}xf(x)=(x3+4x)sin−1x

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Derivatives of Inverse Trigonometric Functions practice set

Find the derivative of the given function.
$f(x)=(x^3+4x)\(\sin\)^{-1}x$