Question

find derivative of x to the power y

Answer 1

assume that x^y=z then differentiate with respect to x.Then the above exp. comes out .Be free and ask if any doubt

Answer 2

Concept:

The derivative is that the instantaneous rate of change of a function with relevancy one in all its variables. this can be love finding the slope of the tangent line to the function at some extent.

Given:

The given function is $x^{y}$.

Find:

We have to seek out the derivative of the given function.

Solution:

Firstly, we'll find the derivative  $x^{y}$ with relevancy to $x$.

Let us assume $z=x^y$.

Now, we are going to take log on both sides, we get

$\log z=y\log x$

Further, we'll differentiate either sides with respect to $x,$ we get

$\begin{aligned}\frac{d}{dx}(\log z)=\frac{d}{dx}(y\log x)\end$

As we know, derivative of $\log x$ is equal to $\frac{1}{x}$ and also use the derivative product rule, we get

$\begin{aligned}\frac{1}{z}\frac{dz}{dx}&=\frac{1}{x}y+\log x\frac{dy}{dx}\end$

Furthermore, we are going to simplify the above expression, we get

$\frac{dz}{dx}=\left(\frac{y}{x}+\log x\frac{dy}{dx}\right)z$

Now, substitute the worth of $z=x^y$ within the above expression, we get

$\frac{dz}{dx}=x^{y}\left(\frac{y}{x}+\log x \frac{dy}{dx}\right)$

Hence, the derivative of $x^y$ is $\frac{dz}{dx}=x^{y}\left(\frac{y}{x}+\log x \frac{dy}{dx}\right)$.

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