Question

If ${{x}^{2}}+{{y}^{2}}=25$ and $\dfrac{dy}{dt}=6$, how do you find $\dfrac{dx}{dt}$ when $y=4$ ?

Answer 1

In order to find a solution to this problem, we will use implicit differentiation. For implicit differentiation, the process will go like differentiating all terms with respect to $time$, use derivative rules and then solve for $\dfrac{dx}{dt}$.

Complete step by step solution:
From the above problem as we can see that the function cannot be solved explicitly, so we will use implicit differentiation.
Implicit differentiation is used when the function cannot be solved explicitly.
We have our expression as:
${{x}^{2}}+{{y}^{2}}=25$
By differentiating our expression with respect to time, we get:
$\dfrac{d}{dt}\left( {{x}^{2}}+{{y}^{2}} \right)=\dfrac{d}{dt}\left( 25 \right)$
$2x\dfrac{dx}{dt}+2y\dfrac{dy}{dt}=0$
Now by dividing $2x$ on both side, we get:
$2x\times \dfrac{1}{2x}\times \dfrac{dx}{dt}+2y\times \dfrac{1}{2x}\times \dfrac{dy}{dt}=0\times \dfrac{1}{2x}$
On Simplifying, we get:
$\dfrac{dx}{dt}+\dfrac{y}{x}\dfrac{dy}{dt}=0$
Now by subtracting $\dfrac{y}{x}$ on both the side, we get:
$\Rightarrow \dfrac{dx}{dt}=-\dfrac{y}{x}\dfrac{dy}{dt}\to \left( 1 \right)$
Now, substituting $\dfrac{dy}{dt}=6$ in equation $\left( 1 \right)$,
$\Rightarrow \dfrac{dx}{dt}=-6\dfrac{y}{\sqrt{25-{{y}^{2}}}}\to \left( 2 \right)$
Therefore, we get the rate of change in $x$ as a function of $x$ and $y$. But$x$ and $y$are related by the given equation: ${{x}^{2}}+{{y}^{2}}=25$
Eliminating $x$ and thus expressing $\dfrac{dx}{dt}$ as a function of $y$ only.
Now as we have all the terms, we now have to substitute in equation (2).
That is, we have $y=4$
$\Rightarrow \dfrac{dx}{dt}=-6\dfrac{y}{\sqrt{25-{{y}^{2}}}}$
Now on substituting $y=4$we get:
$\Rightarrow \dfrac{dx}{dt}=-6\dfrac{4}{\sqrt{25-{{4}^{2}}}}$
On simplifying,
$\Rightarrow \dfrac{dx}{dt}=-6\dfrac{4}{\sqrt{25-16}}$
$\Rightarrow \dfrac{dx}{dt}=-6\dfrac{4}{\sqrt{9}}$
On taking root of $\sqrt{9}=3$, we have:
$\Rightarrow \dfrac{dx}{dt}=-6\dfrac{4}{3}$
Therefore, on further simplification, we get:
$\Rightarrow \dfrac{dx}{dt}=-8$
Therefore, $\dfrac{dx}{dt}$ when $y=4$in expression ${{x}^{2}}+{{y}^{2}}=25$ is:
$\Rightarrow \dfrac{dx}{dt}=-8$
Therefore,
$\Rightarrow \dfrac{dx}{dt}=-8$ is the required solution.

Note:
Implicit differentiation is useful when it is difficult to manipulate the equation such that $y$is isolated on one side of the equation.
To differentiate implicitly, we have to differentiate both the sides of the equation. Then, we have to use relevant derivative rules on all terms and then solve for $\dfrac{dy}{dx}$.
Also, while doing derivatives we have to be sure whether from which term we have to derivate. We got confused about which term we have to derivate, so we have to be careful about that.