Determine the derivative of f(x) = x^2 \ln(\cos x).

Question:

Determine the derivative of {eq}f(x) = x^2 \ln(\cos x). {/eq}

Combining Differentiation Rules

We have a collection of techniques that allow us to differentiate functions. Depending on how a function is expressed, we may need to use different rules to find the derivative. Examining the function to determine the procedure that will best lead to the derivative involves choosing between rules such as the Product Rule, Quotient Rule, and Chain Rule, and also determining the order in which to apply them.

The function given to us is the product of two expressions. We have a monomial multiplied by a logarithmic expression. This logarithmic expression is also a composite expression, as the piece inside of the logarithm is a trigonometric function. Thus, we will need to use both the Product Rule and the Chain Rule to find this derivative. We will use the Product Rule as the overall method for differentiation, and the Chain Rule when defining the pieces that are required in the Product Rule.

First, defining the pieces:

{eq}\begin{align*} &g = x^2 && g' = 2x\\ &h = \ln (\cos x) && h' = \frac{1}{\cos x} \cdot -\sin x = -\tan x \end{align*} {/eq}

Let's now combine this information in order to determine the derivative of this function.

{eq}\begin{align*} f' &= g'h+gh'\\ &= 2x \ln (\cos x) + x^2 (-\tan x)\\ &= 2x \ln (\cos x) - x^2 \tan x \end{align*} {/eq}