# Whats \frac{d}{dx}\ \frac{d}{dy}sin(4x^(\frac{5}{2})z^2)^*cos(in(y^2))? explain the notation

## Question:

Whats {eq}\frac{d}{dx}\ \frac{d}{dy}sin(4x^(\frac{5}{2})z^2)^*cos(ln(y^2))? {/eq} explain the notation

## Partial Derivative:

The derivative with respect to one variable of the multivariable function is known as the partial derivative. Also, we can use the chain rule to find the derivative of the multivariable function.

In the problem, we have the find the partial derivative, as we are given the d-symbol.

So we have the expression:

{eq}\frac{d}{dx}\ \frac{d}{dy}sin(4x^(\frac{5}{2})z^2)^*cos(ln(y^2)) {/eq}

The notation:

{eq}\frac{d}{dx}\ \frac{d}{dy}sin(4x^(\frac{5}{2})z^2)^*cos(ln(y^2)) {/eq}

means that we will first find the partial derivative of the expression:

{eq}sin(4x^(\frac{5}{2})z^2)^*cos(ln(y^2)) {/eq} with respect to y and then the result is differentiated partially with respect to x.

using the expression: {eq}\frac{d}{dx} {/eq}

Now, in order to solve the expression:

{eq}\frac{d}{dx}\left(\frac{d}{dy}\sin \left(4x^{\frac{5}{2}}z^2\right)\cdot \cos \left(\ln \left(y^2\right)\right)\right)\\ {/eq}

So we solve

{eq}\frac{d}{dy}\sin \left(4x^{\frac{5}{2}}z^2\right)\cdot \cos \left(\ln \left(y^2\right)\right)\\ =\sin \left(4x^{\frac{5}{2}}z^2\right)\left(-\sin \left(\ln \left(y^2\right)\right)\right)\frac{2}{y}~~~~~~~~~~~~~~~~~~~~~~~~~~\left [ \because \frac{d}{du}\left(\cos \left(u\right)\right)=-\sin \left(u\right)~~and~~~\frac{d}{dy}\left(\ln \left(y^2\right)\right)=\frac{2}{y} \right ]\\ =-\frac{2\sin \left(4z^2x^{\frac{5}{2}}\right)\sin \left(\ln \left(y^2\right)\right)}{y}\\ {/eq}

Now we have to find the partial derivative wrt x, as follows:

{eq}\frac{d}{dx}\left(-\frac{2\sin \left(4z^2x^{\frac{5}{2}}\right)\sin \left(\ln \left(y^2\right)\right)}{y}\right)\\ =-\frac{2\sin \left(\ln \left(y^2\right)\right)}{y}\cos \left(4z^2x^{\frac{5}{2}}\right)\cdot \:10z^2x^{\frac{3}{2}}~~~~~~~~~~~~~~~~~~~~~~~\left [ \because \frac{d}{du}\left(\sin \left(u\right)\right)=\cos \left(u\right)~~~and~~~~~\frac{d}{dx}\left(x^a\right)=a\cdot x^{a-1} \right ]\\ =-\frac{20z^2x^{\frac{3}{2}}\cos \left(4z^2x^{\frac{5}{2}}\right)\sin \left(\ln \left(y^2\right)\right)}{y} {/eq}