Calculate the derivative of the following functions.
y = sin (4x³ + 3x +1)

Verified step by step guidance

Step 1: Identify the outer function and the inner function. Here, the outer function is \( \sin(u) \) and the inner function is \( u = 4x^3 + 3x + 1 \).

Step 2: Apply the chain rule for differentiation, which states that \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

Step 3: Differentiate the outer function \( \sin(u) \) with respect to \( u \). The derivative is \( \cos(u) \).

Step 4: Differentiate the inner function \( u = 4x^3 + 3x + 1 \) with respect to \( x \). The derivative is \( 12x^2 + 3 \).

Step 5: Combine the results from steps 3 and 4 using the chain rule: \( \frac{dy}{dx} = \cos(4x^3 + 3x + 1) \cdot (12x^2 + 3) \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and chain rule.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is essential when differentiating functions that are nested within each other, such as the sine function in the given problem.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate angles to the ratios of sides in right triangles. In calculus, these functions are important because they have specific derivatives: for example, the derivative of sin(x) is cos(x). Understanding how to differentiate these functions is crucial when working with problems involving trigonometric expressions, as seen in the given function.

Recommended video:

6:04

Introduction to Trigonometric Functions

Related Practice

Textbook Question

A feather dropped on the moon On the moon, a feather will fall to the ground at the same rate as a heavy stone. Suppose a feather is dropped from a height of 40 m above the surface of the moon. Its height (in meters) above the ground after t seconds is s = 40−0.8t². Determine the velocity and acceleration of the feather the moment it strikes the surface of the moon.

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

c. It is impossible for the instantaneous velocity at all times a≤t≤b to equal the average velocity over the interval a≤t≤b.

Textbook Question

Derivatives and tangent lines

b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.

f(x) = 1/ √x; a= 1/4

Textbook Question

Find the derivative function f' for the following functions f.

f(x) = √3x+1; a=8

views

Textbook Question