Video: Differentiate the Composition of a Trigonometric Function and a Logarithmic Function

Video Transcript

Differentiate the function 𝑦 is equal to the tan of the natural logarithm of 𝑎 𝑥 plus 𝑏.

The question is asking us to differentiate the composition of three functions. We have an innermost linear function, then we have a logarithmic function, and finally we have a trigonometric function. We can differentiate the composition of functions by using the chain rule.

The chain rule tells us that if 𝑦 is a function of 𝑢 and 𝑢 is a function of 𝑥, then the derivative of 𝑦 with respect to 𝑥 is equal to the derivative of 𝑦 with respect to 𝑢 multiplied by the derivative of 𝑢 with respect to 𝑥. So, if we set our function 𝑢 to be the natural logarithm of 𝑎𝑥 plus 𝑏, we see that 𝑦 is equal to the tangent of 𝑢. It’s now a function of 𝑢. And 𝑢 is a function of 𝑥. So, we can apply the chain rule.

The chain rule tells us the derivative of 𝑦 with respect to 𝑥 is equal to the derivative of 𝑦 with respect to 𝑢 multiplied by the derivative of 𝑢 with respect to 𝑥. At this point, we can calculate the derivative of 𝑦 with respect to 𝑢. However, the derivative of 𝑢 with respect to 𝑥 is itself the derivative of the composition of two functions. So, we’ll need to apply the chain rule again.

We’ll start by setting 𝑣 equal to the inner linear function 𝑎𝑥 plus 𝑏. This gives us that 𝑢 is equal to the natural logarithm of 𝑣. 𝑢 is a function of 𝑣. And 𝑣 is a function of 𝑥. This means we can calculate the derivative of 𝑢 with respect to 𝑥 by using the chain rule. The derivative of 𝑢 with respect to 𝑥 is equal to the derivative of 𝑢 with respect to 𝑣 multiplied by the derivative of 𝑣 with respect to 𝑥.

Substituting this into our expression for the derivative of 𝑦 with respect to 𝑥 gives us that d𝑦 d𝑥 is equal to d𝑦 by d𝑢 times d𝑢 by d𝑣 times d𝑣 by d𝑥. We’re now ready to start evaluating this expression. We’ll start by d𝑦 by d𝑢, which is the derivative of the tan of 𝑢 with respect to 𝑢. We know the derivative of the tan of 𝜃 with respect to 𝜃 is equal to the sec squared of 𝜃. So, the derivative of the tan of 𝑢 with respect to 𝑢 is the sec squared of 𝑢.

Now, we need to calculate d𝑢 by d𝑣. That’s the derivative of the natural logarithm of 𝑣 with respect to 𝑣. And we know the derivative of the natural algorithm of 𝑥 with respect to 𝑥 is equal to one over 𝑥. So, the derivative of the natural logarithm of 𝑣 with respect to 𝑣 is one over 𝑣. Finally, we need to find d𝑣 by d𝑥. That’s the derivative of 𝑥𝑎 plus 𝑏 with respect to 𝑥. The derivative of 𝑎𝑥 is just 𝑎. And the derivative of a constant 𝑏 is zero. So, this evaluates to give us 𝑎. Multiplying all of these together gives us 𝑎 times the sec squared of 𝑢 all divided by 𝑣.

Finally, we want to rewrite our answer in terms of 𝑥. So, we use that 𝑢 is equal to the natural logarithm of 𝑎𝑥 plus 𝑏 and 𝑣 is equal to 𝑎𝑥 plus 𝑏 to rewrite our answer as 𝑎 times the sec squared of the natural logarithm of 𝑎𝑥 plus 𝑏 all divided by 𝑎𝑥 plus 𝑏. Therefore, we’ve shown the derivative of the function 𝑦 is equal to the tan of the natural logarithm of 𝑎𝑥 plus 𝑏 is equal to 𝑎 times the sec squared of the natural logarithm of 𝑎𝑥 plus 𝑏 all divided by 𝑎𝑥 plus 𝑏.