Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ h(p)=\left(p^{3}-2\right)^{2} $$

Short answer

The derivative of the function \(h(p) = (p^{3}-2)^{2}\) is \(h'(p) = 6p^{2}(p^{3} - 2)\). The chain rule of differentiation was used to find the derivative.

Step-by-step solution

Identify the outer and inner function.

Here, we identify that the function \(h(p)\) is a composite function. The outer function is \(u(v) = v^{2}\) and the inner function is \(v = p^{3} - 2\). We will differentiate each separately.

Apply the chain rule and find derivative of \(v\).

The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. So, first, we differentiate \(v\) with respect to \(p\) using the power rule. The rule states that if you have a function \(f(x) = x^n\), its derivative \(f'(x) = nx^{n-1}\) . Hence, \(dv/dp = 3p^{2}\).

Apply the chain rule and find derivative of \(u\).

Next, find the derivative of \(u\) with respect to \(v\), which also utilises the power rule. Hence, \(du/dv = 2v\).

Multiply the derivatives using chain rule.

Finally, substitute back into the chain rule formula \(du/dp = (du/dv) * (dv/dp)\). Therefore, \(h'(p) = 2v * 3p^{2} = 2(p^{3}-2) * 3p^{2} = 6p^{2}(p^{3} - 2).\

Key concepts

Chain Rule

The chain rule is an essential technique in calculus for differentiating composite functions. When you have a function within another function, you need the chain rule to find the derivative effectively.

• Composite Functions: These are functions where one function is applied to the result of another, such as \(h(p)=\left(p^{3}-2\right)^{2}\).
• How It Works: The chain rule tells you that to find the derivative of such functions, you need to multiply the derivative of the outer function by the derivative of the inner function.

In our exercise, we identified the outer function as \(v^2\) and the inner function as \(p^3 - 2\). By applying the chain rule, we derive \(du/dp = (du/dv) \times (dv/dp)\). This is crucial for correctly computing the derivative of complex functions.

Power Rule

The power rule is one of the simplest and most frequently used differentiation rules. It makes finding derivatives of power functions straightforward.

• Power Functions: A power function is any function of the form \(f(x) = x^n\), where \(n\) is a real number.
• Applying the Power Rule: To differentiate, move the exponent down as a coefficient and subtract one from the exponent. This gives \(f'(x) = nx^{n-1}\).

In our exercise, we used the power rule in two steps. For the inner function \(v = p^3 - 2\), the power rule helps compute \(dv/dp = 3p^2\). Similarly, for the outer function \(u(v) = v^2\), the derivative is \(du/dv = 2v\). By using the power rule, we break down complex problems into simpler parts to enhance understanding and ease of computation.

Composite Function

Composite functions are at the heart of many differentiation problems and involve combining two or more functions into one. Whether you realize it or not, composite functions are everywhere in calculus.

• Understanding Composition: Think of composite functions like nesting dolls — one function living inside another.
• Example: In the function \(h(p)=\left(p^{3}-2\right)^{2}\), \(p^3 - 2\) is the inner function and \(v^2\) is the outer one, showing the composite nature.

Composite functions are crucial for applying differentiation rules like the chain rule effectively. By recognizing a composite function, you can strategically deconstruct a derivative computation into more manageable pieces, improving both learning and computational efficiency. This recognition allows for proper application of differentiation techniques, leading to more robust problem-solving skills in calculus.