As we saw in Chapter 15 (The Derivative including the Power Rule), the Power Rule will determine the derivative of every polynomial function. However, not every function is a polynomial. Other functions exist that are exponential, logarithmic, trigonometric, and combinations of these functions. For example, sometimes we need to differentiate functions such as: $\sin x$, $\cos x$, $\ln x$, $\mathrm e^x$, $\ln(\cos x)$, or $\mathrm e^{\cos x}$.

Without going into the formal proofs of these derivatives, the following rules have been established for differentiating these types of functions. These rules for differentiation will be given to you when you write your IB papers for the A&A SL course.

$\dfrac{\mathrm d}{\mathrm dx}(\sin x)=\cos x$
$\dfrac{\mathrm d}{\mathrm dx}(\cos x)=−\sin x$
$\dfrac{\mathrm d}{\mathrm dx}(\mathrm e^x) = \mathrm e^x$
$\dfrac{\mathrm d}{\mathrm dx}(\ln x)=1x$ 

Oftentimes we are interested in differentiating functions that are composition functions. That is to say, there is one function “inside” another function. To differentiate these types of functions, we use what is called The Chain Rule. The Chain Rule states:

Given a function of the form $y=f(g(x))$ then $\dfrac{\mathrm d}{\mathrm dx}f(g(x))=f'(g(x))\cdot g'(x)$

Many students remember this rule as “The derivative of the outside function times the derivative of the inside function”, where the outside function is $f(x)$ and the inside function is $g(x)$. Lets consider a few examples.

Example 1: $y=\sin(x^2)$. Determine $\dfrac{\mathrm dy}{\mathrm dx}$

Explanation: It is important to understand that this function is a composition of two functions with the “outside” function $f(x)= \sin x$ and the “inside” function.

$g(x)=x^2$. Therefore, since $\dfrac{\mathrm d}{\mathrm dx}(\sin x)=\cos x$ and $\dfrac{\mathrm d}{\mathrm dx}(x^2)=2𝑥$, then $\dfrac{\mathrm dy}{\mathrm dx} = \cos(x^2)\cdot 2x=2x \cos(x^2)$.

Example 2: $y=\mathrm e^{4\cos x}$. Determine $\dfrac{\mathrm dy}{\mathrm dx}$ 

Explanation: Again, it is important to understand that this function is a composition of two functions with the “outside” function $f(x)=e^x$ and the “inside” function $g(x)=4\cos x$. Therefore, since $\dfrac{\mathrm d}{\mathrm dx}(\mathrm e^x)=\mathrm e^x$ and $\dfrac{\mathrm d}{\mathrm dx}(4\cos x)=−4\sin x$, then $\dfrac{\mathrm dy}{\mathrm dx} = \mathrm e^{4\cos x} \cdot (−4sin𝑥)=−4\sin x \mathrm e^{4\cos x}$.