Question

How do you differentiate ${\sin ^4}\left( x \right)$?

Answer 1

The above question is based on the concept of differentiation of trigonometric functions. The main approach to solve this problem we have to first identify whether it is a composite function and if it is a composite function, we can derive the outer function and then the inner function.

Complete step by step solution:
The basic trigonometric functions include six functions i.e., sine, cosine, tangent, cotangent, secant and cosecant.
All of these functions are continuously differentiable in their domain.
Here differentiation of sine function will give cosine function.
So above given is a trigonometric function,
${\sin ^4}(x)$
The above function can be called a composite trigonometric function. Composite functions is a function contained inside another function. In general, if there are two functions f(x) and g(x) then the composite function will be f[g(x)]. i.e., g(x) is inside the f(x) function.
So, in the above question we have sine as outer function and x as inner function.
Now according to the chain rule, when we have a function inside another function it can be differentiable in the following way.
Let x be u, therefore ${\sin ^4}\left( u \right)$
Mathematically this becomes,
$\dfrac{d}{{dx}}{\sin ^4}\left( u \right) = 4{\cos ^3}\left( u \right) \times \dfrac{d}{{dx}}\left[ u \right]$
So, we can see that derivative of sine function is cosine and the power is reduced and derivative of inner function is multiplied with it.

$$\dfrac{d}{{dx}}{\sin ^4}(u) = 4{\cos ^3}(u) \times 1 \\\ \therefore \dfrac{d}{{dx}}\sin \left( x \right) = 4{\cos ^3}x \\\$$

Therefore, we get the above solution.

Note: An important thing to note is that the derivative of a term with power 1 will always result in zero. In the above solution when we derive the term u which has a single power of 1 so on, derivating the power reduces to 0 and leaves the number 1.