Derivative of tanx with respect to sinx

The derivative of tanx with respect to sinx is equal to sec³x, and it is denoted by d/dsinx (tanx). That is,

$\dfrac{d}{d \sin x}(\tan x)=\sec^3 x$.

Derivative of tanx w.r.t sinx

Question: Find the derivative of tanx w.r.t sinx, that is,

Find $\dfrac{d}{d \sin x}(\tan x)$.

Answer:

Let us put

u=tanx and v=sinx.

Here we need to find $\dfrac{du}{dv}$. Differentiating u and v with respect to x, we get that

$\dfrac{du}{dx}=\sec^2 x$ and $\dfrac{dv}{dx}=\cos x$

Now, we have that

$\dfrac{du}{dv} = \dfrac{\frac{du}{dx}}{\frac{dv}{dx}}$

⇒ $\dfrac{du}{dv}$ = $\dfrac{\sec^2 x}{\cos x}$

⇒ $\dfrac{du}{dv}$ = sec²x ⋅ secx, as we know that secx =1/cosx.

⇒ $\dfrac{du}{dv}$ = sec³x.

So the derivative of tanx with respect to sinx is equal to sec³x.

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