Derivatives and Rates of Change

A reliable method

1. Identify how the function is built.
2. Apply the appropriate derivative rule to each part.
3. For a nested function multiply by the inner derivative.
4. Simplify and distinguish the derivative from a numerical derivative value at one input.

Example 1

Differentiate \(f(x)=3 x^{4} - 2 x + 7\) with respect to x.

1. Identify sums, products, and nested functions before choosing a derivative rule.
2. Differentiate each additive term, using the power rule and chain rule where needed: \(\frac{d}{d x} 7\) = \(0\), \(\frac{d}{d x} \left(- 2 x\right)\) = \(-2\), \(\frac{d}{d x} 3 x^{4}\) = \(12 x^{3}\).
3. Add the derivatives and simplify: \(12 x^{3} - 2\). The derivative is a rate function, not the original function value.

Answer: \(12 x^{3} - 2\)

Example 2

Differentiate \(f(x)=\left(2 x + 1\right)^{3}\) with respect to x.

1. Identify sums, products, and nested functions before choosing a derivative rule.
2. Differentiate each additive term, using the power rule and chain rule where needed: \(\frac{d}{d x} 1\) = \(0\), \(\frac{d}{d x} 6 x\) = \(6\), \(\frac{d}{d x} 8 x^{3}\) = \(24 x^{2}\), \(\frac{d}{d x} 12 x^{2}\) = \(24 x\).
3. Add the derivatives and simplify: \(6 \left(2 x + 1\right)^{2}\). The derivative is a rate function, not the original function value.

Answer: \(6 \left(2 x + 1\right)^{2}\)

Common mistakes

• The power rule lowers the exponent by 1 and multiplies by the old exponent.
• A chain rule calculation is incomplete without the inner derivative.

Practice without the answer first

1. Differentiate \(f(x)=5 x^{2} + 4 x - 9\) with respect to x.
2. Differentiate \(f(x)=x^{5} - 3 x^{3}\) with respect to x.
3. Differentiate \(f(x)=\left(x^{2} + 1\right)^{2}\) with respect to x.

These original skill exercises are not official exam questions or a score prediction. Use the coverage notes on your exam hub to choose relevant practice.