91Ó°ÊÓ

The power rule is one of the essential tools in calculus for computing derivatives. It is particularly handy when dealing with polynomials. The power rule states that if you have a function in the form of \(x^n\), the derivative is \(n \cdot x^{n-1}\). This means you multiply the power by the coefficient and then decrease the power by one.

When using the power rule:

• Identify the exponent \(n\) of your function \(x^n\).
• Multiply \(n\) by any constant in front of \(x\).
• Reduce the exponent by one to get the new power for \(x\).

For instance, in the exercise example, when calculating the derivative of \(h(x) = 5x^{-3}\), we apply the power rule with \(n = -3\), resulting in a derivative of \(h'(x) = -15x^{-4}\). By simply following these steps, you can effectively use the power rule on similar expressions.

The chain rule is a powerful method used in calculus to find the derivative of composite functions. A composite function is when one function is nested inside another, as in the example \(y = 5(7x^3 + 1)^{-3}\). The chain rule helps us differentiate such functions by linking them through their individual derivatives.

The core idea of the chain rule is as follows:

• Find the derivative of the outer function, keeping the inner function intact.
• Multiply this result by the derivative of the inner function.

In the exercise, after finding the derivatives of \(h(x)\) and \(g(x)\), we apply the chain rule. We first differentiate the outer function \(h\), evaluated at \(g(x)\), and then multiply it by \(g'(x)\). This results in \(y'(x) = -15(7x^3 + 1)^{-4} \cdot 21x^2\). By understanding the chain rule, you can handle more complex derivatives with confidence.

Composite functions involve applying one function inside another. They are prevalent in calculus and require systematic approaches to tackle, especially when computing derivatives.

To handle composite functions:

• Identify the outer function and the inner function in the composite.
• Derive each function individually if needed, starting with the innermost.
• Use rules like the chain rule to combine these derivatives correctly.

In the exercise, the composite function is \(y = 5(7x^3 + 1)^{-3}\). The outer function \(h(x) = 5x^{-3}\) and the inner function \(g(x)=7x^3+1\) are clear upon inspection. By breaking them apart, applying the power rule, and using the chain rule, we simplify the task of finding the complete derivative. Mastering the decomposition of composite functions can greatly simplify complex calculus problems.