Derivative of mod sinx | Mod sinx Derivative

The derivative of the modulus of sinx is equal to (sinx cosx)/|sinx|. In this post, we will learn how to differentiate mod sinx, that is, how to find d/dx(|sinx|).

Mod sinx Derivative Formula

The derivative of mod sinx is given below:

$\dfrac{d}{dx}(|\sin x|)=\dfrac{\sin x \cos x}{|\sin x|}$, provided that sinx is non-zero.

Before proving this formula, let us recall the derivative of mod x which is given below:

$\dfrac{d}{dx}(|x|)=\dfrac{x}{|x|}$ for $x \neq 0$ …(I)

Derivative of modulus of sinx

Question: Find $\dfrac{d}{dx}(|\sin x|)$.

Answer:

We will use the cahin rule of derivatives and the above formula (I) to find the derivative of mod sinx. Let us put

z=sinx.

⇒ dz/dx = cosx …(II)

Then by the chain rule,

$\dfrac{d}{dx}(|\sin x|)$ = $\dfrac{d}{dz}(|z|) \times \dfrac{dz}{dx}$

= $\dfrac{z}{|z|} \times \cos x$ by (I) and (II)

= $\dfrac{\sin x \cos x}{|\sin x|}$ as z=sinx.

So the derivative of mod sinx is (sinx cosx)/|sinx| and this is obtained by the chain rule of derivatives.

Question 1: Find the derivative of mod sin2x.

Answer:

Let t=2x.

⇒ dt/dx = 2

By the chain rule of derivatives,

$\dfrac{d}{dx}(|\sin 2x|)$ = $\dfrac{d}{dt}(|\sin t|) \times \dfrac{dt}{dx}$

= $\dfrac{\sin t \cos t}{|\sin t|} \times 2$ by the above formula.

= $\dfrac{2\sin t \cos t}{|\sin t|}$

= $\dfrac{\sin 2t}{|\sin t|}$ using the formula sin2x=2sinx cosx.

= $\dfrac{\sin 4x}{|\sin 2x|}$ as t=2x.

So the derivative of mod sin2x is equal to sin4x/|sin2x|.

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