Question

How do you find the derivative of $y=\dfrac{5}{{{(x-3)}^{2}}}$ ?

Answer 1

We start solving the given question $y=\dfrac{5}{{{(x-3)}^{2}}}$ by using the chain rule $\dfrac{dy}{dx}=\left( \dfrac{dy}{dz} \right)\left( \dfrac{dz}{dx} \right)$ in the given equation. Here, we can take z as x - 3. Then we can solve the derivatives separately. And finally multiply both the derivatives to get the final answer.

Complete step by step solution:
According to the problem, we are asked to find the derivative of the equation $y=\dfrac{5}{{{(x-3)}^{2}}}$--- ( 1 )
Therefore, by using the chain rule $\dfrac{dy}{dx}=\left( \dfrac{dy}{dz} \right)\left( \dfrac{dz}{dx} \right)$, in equation 1, let us do the following:
Let us take z = x - 3
First, we calculate for $\dfrac{dy}{dz}$. Now, let us put z = x – 3 in $\dfrac{dy}{dz}$.
Therefore, $\dfrac{dy}{dz}=\dfrac{d(\dfrac{5}{{{z}^{2}}})}{dz}$ [Since z = x - 3]
$\Rightarrow \dfrac{dy}{dz}=\dfrac{d(5.z^{-2})}{dz}$
Let us use the formula - $\dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}$. So, by doing this, we get the following.
$\Rightarrow \dfrac{dy}{dz}=\left(5\right)\left(-2\right){{z}^{-3}}$
$\Rightarrow \dfrac{dy}{dz}=-10{{z}^{-3}}$ ----(2)
Let us consider the above equation as 2.
Now, let us calculate for $\dfrac{dz}{dx}$.
Let us again put z = x - 3, in the above equation, that is $\dfrac{dz}{dx}$. Therefore, we get
$\Rightarrow \dfrac{dz}{dx}=\dfrac{d\left(x-3\right)}{dx}$
$\Rightarrow \dfrac{dz}{dx}=\dfrac{d\left(x\right)}{dx}-\dfrac{d\left(3\right)}{dx}$
$\Rightarrow \dfrac{dz}{dx}=1-0$
$\Rightarrow \dfrac{dz}{dx}=1$ ----(3)
Since, we calculated both $\dfrac{dz}{dx}$ and $\dfrac{dy}{dz}$, let us substitute both the equations (2) and (3) in equation (1), that is $\dfrac{dy}{dx}=\left( \dfrac{dy}{dz} \right)\left( \dfrac{dz}{dx} \right)$.
$\Rightarrow \dfrac{dy}{dx}=(-10{{z}^{-3}})(1)$ ----(4)
But now, we should substitute, z = x - 3. Therefore, let us substitute equation (4) with z = x - 3.
$\Rightarrow \dfrac{dy}{dx}=(-10{(x-3)}^{-3})$
$\Rightarrow \dfrac{dy}{dx}=\dfrac{-10}{{{(x-3)}^{3}}}$
So, we have found the derivative of the given equation $y=\dfrac{5}{{{(x-3)}^{2}}}$ as $\dfrac{dy}{dx}=\dfrac{-10}{{{(x-3)}^{3}}}$
Therefore, the solution of the given equation $y=\dfrac{5}{{{(x-3)}^{2}}}$ is $\dfrac{dy}{dx}=\dfrac{-10}{{{(x-3)}^{3}}}$

Note:
Whenever we get this type of problem, we should always be careful while substituting the equations.Also, we could also find the derivative of the equation using the quotient rule which is $y'=\dfrac{f'(x)g(x)-f(x)g'(x)}{{{(g(x))}^{2}}}$ . You could also check the answer by integrating the answer to get back the question.