Question

Compute the derivative of the given function. $$f(x)=\left(4 x^{3}-x\right)^{10}$$

Short answer

The derivative is \( f'(x) = 10(4x^3 - x)^9 \cdot (12x^2 - 1) \).

Step-by-step solution

Identify the Outer Function and Inner Function

The given function is a composition, so we identify the outer and inner functions for differentiation using the chain rule. Here, the outer function is \( g(u) = u^{10} \) and the inner function is \( u = 4x^3 - x \).

Differentiate the Outer Function

Differentiate the outer function \( g(u) = u^{10} \) with respect to \( u \). Using the power rule, the derivative is \( g'(u) = 10u^9 \).

Differentiate the Inner Function

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

Apply the Chain Rule

Apply the chain rule: \( f'(x) = g'(u) \cdot u'(x) \). Substitute \( g'(u) = 10u^9 \) and \( u'(x) = 12x^2 - 1 \).

Substitute Back the Inner Function

Substitute back the inner function \( u = 4x^3 - x \) into the derivative to get the final expression: \( f'(x) = 10(4x^3 - x)^9 \cdot (12x^2 - 1) \).

Simplify the Expression

The expression simplifies to \( f'(x) = 10(4x^3 - x)^9 \times (12x^2 - 1) \). This is the derivative of the function.

Key concepts

Derivative Calculation

When you come across a mathematical function and are asked to compute its derivative, you are essentially being asked to find the rate at which the function's output changes as its input changes. This process is one of the fundamental tools of calculus. In this exercise, the given function is \[ f(x) = (4x^3 - x)^{10} \] To calculate its derivative, we need to employ the chain rule because it is a composition of an outer function and an inner function. Derivatives tell us how functions change and can help us understand curves, optimize systems, and predict future values. The derivative, noted as \( f'(x) \), reflects the slope or steepness of the function, allowing us to learn how the function behaves when \( x \) changes.

Power Rule

One integral tool for calculating derivatives is the power rule. This rule simplifies the differentiation process for any function of the form \( x^n \). Essentially, the power rule states that the derivative of \( x^n \) is \( nx^{n-1} \).
For example, to compute the derivative of \( u = 4x^3 - x \), you would do the following:

• The derivative of \( 4x^3 \) is computed as \( 3 \times 4x^{3-1} = 12x^2 \)
• The derivative of \( -x \) is \(-1 \)

Thus, the derivative of the inner function \( u \) is \( u'(x) = 12x^2 - 1 \). The power rule makes it efficient to handle polynomial derivatives without unnecessary complexity.

Inner and Outer Functions

When dealing with composite functions, like the one in this exercise, it is crucial to identify both the inner and outer functions. The function \[ f(x) = (4x^3 - x)^{10} \] is a perfect example of this structure. Here, the outer function is a power function \( g(u) = u^{10} \) and, the inner function is a polynomial \( u = 4x^3 - x \).
Identifying these components is essential because it allows us to apply the chain rule efficiently.

• The outer function is differentiated first: \( g'(u) = 10u^9 \).
• The derivative of the inner function tasks you to find how \( u \) changes with respect to \( x \), which is \( 12x^2 - 1 \).

Once both derivatives are found, the chain rule tells us to multiply them to get the overall derivative of the composite function, resulting in \( f'(x) = 10(4x^3 - x)^9 \cdot (12x^2 - 1) \). Understanding inner and outer functions is the cornerstone of effectively applying the chain rule.