Skip to main content
Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.9.23
Chapter 3, Problem 3.9.23

Derivatives in Differential Form


In Exercises 17–28, find dy.


y = sin(5√x)

Verified step by step guidance
1
First, identify the function y = sin(5√x). We need to find the derivative dy/dx.
Recognize that the function involves a composition of functions: the sine function and the square root function. Use the chain rule to differentiate.
The chain rule states that if you have a composite function y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). Here, f(u) = sin(u) and g(x) = 5√x.
Differentiate the outer function f(u) = sin(u) with respect to u, which gives f'(u) = cos(u).
Differentiate the inner function g(x) = 5√x with respect to x. Recall that √x can be expressed as x^(1/2), so g(x) = 5x^(1/2). The derivative g'(x) = 5 * (1/2) * x^(-1/2).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative dy/dx is found by multiplying the derivative of the outer function f with respect to the inner function g by the derivative of the inner function g with respect to x. This is essential for differentiating y = sin(5√x), where the outer function is sine and the inner function is 5√x.
Recommended video:
05:02
Intro to the Chain Rule

Derivative of Sine Function

The derivative of the sine function, sin(x), is cos(x). This basic derivative rule is crucial when applying the chain rule to differentiate y = sin(5√x). After applying the chain rule, the derivative of the outer function, sin, is needed, which transforms into cos when differentiating.
Recommended video:
03:53
Derivatives of Sine & Cosine

Derivative of Square Root Function

The derivative of the square root function, √x, is 1/(2√x). This rule is important when differentiating the inner function 5√x in the given problem. By applying this derivative rule, we can find the rate of change of the inner function, which is necessary for the application of the chain rule.
Recommended video:
01:32
Derivatives of Other Trig Functions Example 1
Related Practice
Textbook Question

Derivative Calculations


In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).


y = √u, u = sin x

Textbook Question

For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.


lim (x → 1) (x⁵⁰ − 1) / (x − 1)

Textbook Question

Second Derivatives


In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.


2√y = x – y

Textbook Question

In Exercises 41–58, find dy/dt.


y = (1 + cos(2t))⁻⁴

Textbook Question

In Exercises 43–50, find by implicit differentiation.


y² = x .

x + 1

Textbook Question

Is there a value of c that will make


f(x) = { (sin²(3x)) / x², x ≠ 0

c, x = 0


continuous at x = 0? Give reasons for your answer.