Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
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 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

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1:36 minutes

Problem 94a

Textbook Question

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.
a. g(1)

Verified step by step guidance

Step 1: Understand the problem. We are given a function g(x) = f(x^2) and need to find g(1). We know that f is differentiable and passes through the point (1, 4), and the tangent line to f at (1, 4) is y = 3x + 1.

Step 2: Evaluate g(1). Since g(x) = f(x^2), we substitute x = 1 into g(x) to get g(1) = f(1^2) = f(1).

Step 3: Use the information about f. We know that the graph of f passes through the point (1, 4), which means f(1) = 4.

Step 4: Conclude the evaluation. Since f(1) = 4, we have g(1) = f(1) = 4.

Step 5: Summarize the result. Therefore, g(1) is equal to 4.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Differentiable Functions

A differentiable function is one that has a derivative at every point in its domain. This means that the function is smooth and continuous, allowing for the calculation of slopes of tangent lines at any point. The existence of a derivative indicates that the function can be locally approximated by a linear function, which is essential for understanding tangent lines.

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. The equation of the tangent line can be derived using the point-slope form, which incorporates the derivative of the function at that point. In this case, the tangent line to f at (1, 4) is given as y = 3x + 1, indicating that the slope at that point is 3.

Composition of Functions

The composition of functions involves combining two functions where the output of one function becomes the input of another. In this problem, g(x) = f(x²) represents a composition where the input x is squared before being passed to function f. Understanding how to evaluate g(1) requires substituting 1 into the composition and then determining the value of f at the resulting input.

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Related Practice

Textbook Question

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (f(f(x))) |x=4

Textbook Question

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (g(f(x))) |x=1

Textbook Question

Composition containing sin x Suppose f is differentiable for all real numbers with f(0)=−3,f(1)=3,f′(0)=3, and f′(1)=5. Let g(x)=sin(πf(x)). Evaluate the following expressions.

b. g'(1)

Textbook Question

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.

c. g'(π)

Textbook Question

{Use of Tech} Hours of daylight The number of hours of daylight at any point on Earth fluctuates throughout the year. In the Northern Hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40° north latitude, the length of a day is approximated by D(t) = 12−3 cos (2π(t+10) / 365), where D is measured in hours and 0≤t≤365 is measured in days, with t=0 corresponding to January 1.

b. Find the rate at which the daylight function changes.

Textbook Question

The Chain Rule for second derivatives

b. Use the formula in part (a) to calculate $\frac{d^{2}}{d x^{2}} (\sin (3 x^{4} + 5 x^{2} + 2))$ .

Textbook Question

Deriving trigonometric identities

a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Textbook Question

Deriving trigonometric identities

b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

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Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.a. g(1)

Key Concepts

Differentiable Functions

Tangent Lines

Composition of Functions

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Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.
a. g(1)