Question

How do you find the derivative of $y=(3{{x}^{2}}+5)(7{{x}^{3}}+8x)$?

Answer 1

We start solving the problem by applying the distributive rule of multiplication that is, $\left( a+b \right)\left( c+d \right)\text{ }=\text{ }ac+ad+bc+bd$ in the given equation. Then you solve the derivatives of the equation separately. Then add all of them in the final derivative of the equation. or you could even use the product rule of differentiation also in the above question.

Complete step by step solution:
According to the problem, we are asked to find the derivative of the equation $y=(3{{x}^{2}}+5)(7{{x}^{3}}+8x)$--- ( 1 )
Therefore, by using the distributive property of multiplication, that is $\left( a+b \right)\left( c+d \right)\text{ }=\text{ }ac+ad+bc+bd$ in equation 1, let us do the following:
$\Rightarrow y=(3{{x}^{2}}+5)(7{{x}^{3}}+8x)$
$\Rightarrow y={3{{x}^{2}}(7x^3)}+5(7{{x}^{3}})+{3{{x}^{2}}(8x)}+5(8x)$
$\Rightarrow y={21x^5}+{35x^3}+{24x^3}+{40x}$
$\Rightarrow y={21x^5}+{59x^3}+{40x}$
Now, let us differentiate both sides with respect to x.
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d21{{x}^{5}}}{dx}+\dfrac{d59{{x}^{3}}}{dx}+\dfrac{d40x}{dx}$---(2)
We divide this equation into three terms. $\dfrac{d21{{x}^{5}}}{dx}$, $\dfrac{d59{{x}^{3}}}{dx}$, $\dfrac{d40x}{dx}$. Now we differentiate term wise.
Let us use the formula - $\dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}$to find the derivative for first term. So, by doing this, we get the following.
$\Rightarrow \dfrac{d21{{x}^{5}}}{dx}=(105){{x}^{4}}$ ---(3)
Let us use the formula - $\dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}$to find the derivative for second term. So, by doing this, we get the following.
$\Rightarrow \dfrac{d59{{x}^{3}}}{dx}=177{{x}^{2}}$ ----(4)
Let us use the formula - $\dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}$to find the derivative for second term. So, by doing this, we get the following.
$\Rightarrow \dfrac{d40x}{dx}=40$ ----(5)
Now let us substitute the equations (3), (4) and (5) in equation (2).
$\Rightarrow \dfrac{dy}{dx}=105{{x}^{4}}+177{{x}^{2}}+40$
So, we have found the derivative of the given equation $y=(3{{x}^{2}}+5)(7{{x}^{3}}+8x)$ as $\dfrac{dy}{dx}=105{{x}^{4}}+177{{x}^{2}}+40$
Therefore, the solution of the given equation $y=(3{{x}^{2}}+5)(7{{x}^{3}}+8x)$ is $\dfrac{dy}{dx}=105{{x}^{4}}+177{{x}^{2}}+40$

Note:
We should be careful while substitutions as if you substitute wrongly, the whole answer could go wrong. Also, we could also find the derivative of the equation using the product rule which is $y'=f'(x)g(x)+f(x)g'(x)$. You could also verify your answer by using integration. After you integrate, if you get the answer as the question, then your answer is correct.