Derivative of x^tanx (x to the power tanx)

The derivative of x^tanx (x to the power tanx) is denoted by d/dx(x^tanx) and its value is equal to x^tanx[tanx/x + sec²x log_ex].

The derivative formula of x^tanx is given by

d/dx(x^tanx) = x^tanx[tanx/x + sec²x log_ex].

Let us now learn how to differentiate x^tanx.

Differentiate x^tanx

Question: Prove that d/dx(x^tanx) = x^tanx[tanx/x + sec²x logx].

Answer:

Let us put

y=x^tanx.

Here we need to find dy/dx. Taking logarithms on both sides, we get that

log_e y = log_e x^tanx

⇒ log_ey = tanx log_ex, as we know the logarithm formula log_abⁿ = n log_ab.

Differentiating both sides w.r.t x, we have

$\dfrac{d}{dx}(\log_e y)=\dfrac{d}{dx}(\tan x \log_e x)$

⇒ $\dfrac{1}{y} \dfrac{dy}{dx}$ $=\tan x\dfrac{d}{dx}(\log_e x)+\log_e x\dfrac{d}{dx}(\tan x)$, by the product rule of derivatives.

⇒ $\dfrac{1}{y} \dfrac{dy}{dx}$ $=\tan x \cdot \dfrac{1}{x}+\log_e x \sec^2 x$ as we know d/dx(log_ex) =1/x and d/dx(tan x)= sec²x.

⇒ $\dfrac{dy}{dx}=y(\dfrac{\tan x}{x}+\sec^2 x\log_e x)$

⇒ $\dfrac{dy}{dx}=x^{\tan x}(\dfrac{\tan x}{x}+\sec^2 x \log_e x)$ as y=x^tanx.

So the derivative of x^tanx (x to the power tanx) is equal to x^tanx[tanx/x + sec²x logx], and it is obtained by the logarithmic differentiation.

More Derivatives:

Derivative of x^x | Derivative of x^sinx

Derivative of x^cosx

Derivative of x^3/2

Derivative of sinx/x | Derivative of x^logx

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