Question

Find the derivative of the following:
$-\dfrac{2651}{504\sqrt[315]{{{x}^{2966}}}}$

Answer 1

Don’t get confused with the large numbers. Use the formula $\dfrac{d}{dx}\left( \dfrac{1}{{{x}^{\dfrac{n}{m}}}} \right)=\dfrac{d}{dx}\left( {{x}^{-\dfrac{n}{m}}} \right)=\dfrac{-n}{m}{{x}^{\dfrac{-n}{m}-1}}$to find the derivative.

Complete step by step solution:
The given function is $-\dfrac{2651}{504\sqrt[315]{{{x}^{2966}}}}$.
Taking out the constant term, we get
\dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{-2651}{504}\times \dfrac{d}{dx}\left\\{ \dfrac{1}{\sqrt[315]{{{x}^{2966}}}} \right\\}
The can be written as,
\dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{-2651}{504}\times \dfrac{d}{dx}\left\\{ \dfrac{1}{{{x}^{\dfrac{2966}{315}}}} \right\\}
So the above equation can be re-written using the formula $\dfrac{d}{dx}\left( \dfrac{1}{{{x}^{\dfrac{n}{m}}}} \right)=\dfrac{d}{dx}\left( {{x}^{-\dfrac{n}{m}}} \right)=\dfrac{-n}{m}{{x}^{\dfrac{-n}{m}-1}},$

& \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{-2651}{504}\times \dfrac{-2966}{315}\times {{x}^{\dfrac{-2966}{315}-1}} \\\ & \Rightarrow \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{-2651}{504}\times \dfrac{-2966}{315}\times {{x}^{\dfrac{-2966-315}{315}}} \\\ & \Rightarrow \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{-2651}{504}\times \dfrac{-2966}{315}\times {{x}^{\dfrac{-3281}{315}}} \\\ & \Rightarrow \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 2966}{504\times 315}\times \dfrac{1}{\sqrt[315]{{{x}^{3281}}}} \\\ \end{aligned}$$ Dividing throughout by ‘2’, we get $$\Rightarrow \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\times \dfrac{1}{\sqrt[315]{{{x}^{3150+131}}}}$$ Now applying the formula $${{x}^{m+n}}={{x}^{m}}.{{x}^{n}}$$ under the root, we have $$\begin{aligned} & \Rightarrow \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\times \dfrac{1}{\sqrt[315]{{{x}^{3150}}.{{x}^{131}}}} \\\ & \Rightarrow \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\times \dfrac{1}{\sqrt[315]{{{x}^{3150}}}\times \sqrt[315]{{{x}^{131}}}} \\\ \end{aligned}$$ Now, by applying the formula $${{x}^{mn}}={{({{x}^{m}})}^{n}}$$ under the root, we get $$\begin{aligned} & \Rightarrow \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\times \dfrac{1}{\sqrt[315]{{{({{x}^{10}})}^{315}}}\times \sqrt[315]{{{x}^{131}}}} \\\ & \Rightarrow \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\times \dfrac{1}{{{x}^{10}}\times \sqrt[315]{{{x}^{131}}}} \\\ \end{aligned}$$ Here we can observe that $(315-131=184)$, so we will rationalise by $$\sqrt[315]{{{x}^{184}}}$$, we get $$\begin{aligned} & \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\dfrac{1}{{{x}^{10}}}\times \dfrac{1}{\sqrt[315]{{{x}^{131}}}}\times \dfrac{\sqrt[315]{{{x}^{184}}}}{\sqrt[315]{{{x}^{184}}}} \\\ & \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\times \dfrac{1}{{{x}^{10}}}\times \dfrac{\sqrt[315]{{{x}^{184}}}}{\sqrt[315]{{{x}^{131}}\times {{x}^{184}}}} \\\ & \Rightarrow \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\times \dfrac{1}{{{x}^{10}}}\dfrac{\sqrt[315]{{{x}^{184}}}}{\sqrt[315]{{{x}^{315}}}} \\\ & \dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\times \dfrac{1}{{{x}^{10}}}\dfrac{\sqrt[315]{{{x}^{184}}}}{x} \\\ \end{aligned}$$ $$\dfrac{d}{dx}\left\\{ \dfrac{-2651}{504\sqrt[315]{{{x}^{2966}}}} \right\\}=\dfrac{2651\times 1483}{252\times 315}\dfrac{\sqrt[315]{{{x}^{184}}}}{{{x}^{11}}}$$ **Note:** In this problem looking at the bigger values the student may get confused. We should always try to solve the problem using a simple basic formula. The student sometimes gets confused to remove the constant term while deriving and will make mistakes.