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

Verified step by step guidance

1. Step 1: Identify the outer function and the inner function. Here, the outer function is \( \sin(u) \) and the inner function is \( u = 4x^3 + 3x + 1 \).
2. Step 2: Apply the chain rule for differentiation, which states that \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).
3. Step 3: Differentiate the outer function \( \sin(u) \) with respect to \( u \). The derivative is \( \cos(u) \).
4. Step 4: Differentiate the inner function \( u = 4x^3 + 3x + 1 \) with respect to \( x \). The derivative is \( 12x^2 + 3 \).
5. Step 5: Combine the results from steps 3 and 4 using the chain rule: \( \frac{dy}{dx} = \cos(4x^3 + 3x + 1) \cdot (12x^2 + 3) \).

Key Concepts

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.