Question

if y =(sinx^3)1/2 then dy/dx

Answer 1

Consider,

$\longrightarrow y = \dfrac{(\sin x^3)}{2}$

We need to differentiate this function w.r.t. (x)

Differentiating both sides will give us:

$\longrightarrow \dfrac{dy}{dx}= \dfrac{d}{dx}\left(\dfrac{\sin x^3}{2}\right)$

$\longrightarrow \dfrac{dy}{dx}=\dfrac{1}{2} \times\dfrac{d}{dx}\left(\sin x^3\right)$

We know that derivative of sin x is cos x. Therefore, by applying chain rule, we get the following result:

$\longrightarrow \dfrac{dy}{dx}=\dfrac{1}{2} \times\left(\cos x^3 \times 3x^2\right)$

$\longrightarrow \dfrac{dy}{dx}=\dfrac{3 x^2 \cos(x^3)}{2}$

Hence the required answer is:

$\underline{\boxed{ \dfrac{d}{dx}\left( \dfrac{\sin x^3}{2}\right) =\dfrac{3 x^2 \cos(x^3)}{2}}}$

Additional Information:

Let's understand what chain rule actually is!

Whenever we are given some composite functions let say f(g(x)). In order to differentiate this function, firstly we need to differentiate the outermost function which is f(x) and then we have to multiply the derivative of f(x) with derivative of g(x). This chain continues for more than 2 functions too.

Let's  know it more by differentiating sin(cos x)).

$\longrightarrow y = \sin(\cos x)$

$\longrightarrow \dfrac{dy}{dx} = \dfrac{d}{dx}(\sin(\cos x))$

Now we know that derivative of sin(x) is cos(x) and derivative of cos(x) is -sin(x). Therefore, to find the derivative of sin(cos(x)), we will differentiate the outermost function first and then inside function.

$\longrightarrow \dfrac{dy}{dx} =\cos(\cos x) \times (- \sin x)$

This is the required derivative.