If {eq}\displaystyle f(x) = \cos(\sin(x^7)) {/eq}, then find {eq}\displaystyle f'(x) {/eq}.

Question:

The Chain Rule Derivative:

In mathematics, the derivative is part of calculus and the derivative of a function is defined as the instantaneous rate of change of the function with respect to its variable. The chain rule of differentiation is a formula that is used to find the derivative of a composite function such as written in the form f\left( {g(x)} \right). The chain rule derivative formula is given by: {eq}\dfrac{d}{{dx}}f\left( {g(x)} \right) = f'\left( {g\left( x \right)} \right) \times g'\left( x \right) {/eq}.

Other derivative formulas that may also help in the given problem are as follows:

{eq}\begin{align*} \dfrac{d}{{dt}}\cos \left( t \right) &= - \sin \left( t \right)\\ \dfrac{d}{{dt}}\sin \left( t \right) &= \cos \left( t \right)\\ \dfrac{d}{{dt}}{t^n} &= n{t^{n - 1}} \end{align*} {/eq}

Answer and Explanation: 1