Question

How do you find the derivative of ${{\sin }^{7}}\left( x \right)$?

Answer 1

Assume the function $\sin x$ as $f\left( x \right)$, so that the given function gets converted into the form ${{\left( f\left( x \right) \right)}^{n}}$ where n = 7. Now, differentiate the function with respect to the value x and use the formula $\dfrac{d\left[ f{{\left( x \right)}^{n}} \right]}{dx}=n\times {{\left( f\left( x \right) \right)}^{n-1}}\times f'\left( x \right)$ where f’(x) is the derivative of the assumed function f(x). Use the basic formula $\dfrac{d\left( \sin x \right)}{dx}=\cos x$ to get the answer.

Complete step by step solution:
Here, we have been provided with the function ${{\sin }^{7}}\left( x \right)$ and we are asked to differentiate it. Let us assume the given function the as y. So, we have,
$\Rightarrow f\left( x \right)=\sin x$
Therefore, assuming the given function as y, we have,
$\Rightarrow y={{\left( f\left( x \right) \right)}^{7}}$
So, we have to differentiate the above function. Clearly, we can see that the above function is of the form $y={{\left( f\left( x \right) \right)}^{n}}$, where n = 7, whose derivative is given by the power reduction formula given as: $\dfrac{d\left[ f{{\left( x \right)}^{n}} \right]}{dx}=n\times {{\left( f\left( x \right) \right)}^{n-1}}\times f'\left( x \right)$, where f’(x) is the derivative of f(x),so using this formula we get on differentiating both the sides with respect to the variable x,
$\begin{aligned} & \Rightarrow \dfrac{dy}{dx}=7\times {{\left( \sin x \right)}^{7-1}}\times \dfrac{d\left[ \sin x \right]}{dx} \\\ & \Rightarrow \dfrac{dy}{dx}=7\times {{\left( \sin x \right)}^{6}}\times \dfrac{d\left[ \sin x \right]}{dx} \\\ & \Rightarrow \dfrac{dy}{dx}=7\times {{\sin }^{6}}x\times \dfrac{d\left[ \sin x \right]}{dx} \\\ \end{aligned}$
Using the basic formula of the derivative of the sine function given as: $\dfrac{d\left( \sin x \right)}{dx}=\cos x$, we get,
$\begin{aligned} & \Rightarrow \dfrac{dy}{dx}=7\times {{\sin }^{6}}x\times \cos x \\\ & \Rightarrow \dfrac{dy}{dx}=7{{\sin }^{6}}x\cos x \\\ \end{aligned}$
Hence, the above relation is our answer.

Note: You must remember all the basic rules and formulas of differentiation like: - power reduction rule, product rule, chain rule, $\dfrac{u}{v}$ rule etc. as they make our question easy to solve. Remember the derivatives of some common functions like: - ${{x}^{n}},{{e}^{x}}$, trigonometric functions, inverse trigonometric functions, logarithmic functions etc. as they are used frequently in calculus.