Question

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

Answer 1

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

Complete step by step solution:
 The given function is $$-{\displaystyle \frac{2651}{504\sqrt[315]{x^{2966}}}}$$.
Taking out the constant term, we get
$${\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{-2651}{504}}\times {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{1}{\sqrt[315]{x^{2966}}}}\right\}$$
The can be written as,
$${\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{-2651}{504}}\times {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{1}{x^{{\displaystyle \frac{2966}{315}}}}}\right\}$$
So the above equation can be re-written using the formula $${\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{1}{x^{{\displaystyle \frac{n}{m}}}}}\right)={\displaystyle \frac{d}{dx}}\left(x^{-{\displaystyle \frac{n}{m}}}\right)={\displaystyle \frac{-n}{m}}{x}^{{\displaystyle \frac{-n}{m}}-1},$$
$$\begin{array}{rl}& {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{-2651}{504}}\times {\displaystyle \frac{-2966}{315}}\times x^{{\displaystyle \frac{-2966}{315}}-1}\\ & \Rightarrow {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{-2651}{504}}\times {\displaystyle \frac{-2966}{315}}\times x^{{\displaystyle \frac{-2966-315}{315}}}\\ & \Rightarrow {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{-2651}{504}}\times {\displaystyle \frac{-2966}{315}}\times x^{{\displaystyle \frac{-3281}{315}}}\\ & \Rightarrow {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 2966}{504\times 315}}\times {\displaystyle \frac{1}{\sqrt[315]{x^{3281}}}}\end{array}$$
Dividing throughout by ‘2’, we get
$$\Rightarrow {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}\times {\displaystyle \frac{1}{\sqrt[315]{x^{3150+131}}}}$$
Now applying the formula $$x^{m+n}=x^m.x^n$$ under the root, we have
$$\begin{array}{rl}& \Rightarrow {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}\times {\displaystyle \frac{1}{\sqrt[315]{x^{3150}.x^{131}}}}\\ & \Rightarrow {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}\times {\displaystyle \frac{1}{\sqrt[315]{x^{3150}}\times \sqrt[315]{x^{131}}}}\end{array}$$
Now, by applying the formula $$x^{mn}={(x^m)}^n$$ under the root, we get
$$\begin{array}{rl}& \Rightarrow {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}\times {\displaystyle \frac{1}{\sqrt[315]{{(x^{10})}^{315}}\times \sqrt[315]{x^{131}}}}\\ & \Rightarrow {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}\times {\displaystyle \frac{1}{x^{10}\times \sqrt[315]{x^{131}}}}\end{array}$$
Here we can observe that $(315-131=184)$, so we will rationalise by $$\sqrt[315]{x^{184}}$$, we get
$$\begin{array}{rl}& {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}{\displaystyle \frac{1}{x^{10}}}\times {\displaystyle \frac{1}{\sqrt[315]{x^{131}}}}\times {\displaystyle \frac{\sqrt[315]{x^{184}}}{\sqrt[315]{x^{184}}}}\\ & {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}\times {\displaystyle \frac{1}{x^{10}}}\times {\displaystyle \frac{\sqrt[315]{x^{184}}}{\sqrt[315]{x^{131}\times x^{184}}}}\\ & \Rightarrow {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}\times {\displaystyle \frac{1}{x^{10}}}{\displaystyle \frac{\sqrt[315]{x^{184}}}{\sqrt[315]{x^{315}}}}\\ & {\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}\times {\displaystyle \frac{1}{x^{10}}}{\displaystyle \frac{\sqrt[315]{x^{184}}}{x}}\end{array}$$
$${\displaystyle \frac{d}{dx}}\left\{{\displaystyle \frac{-2651}{504\sqrt[315]{x^{2966}}}}\right\}={\displaystyle \frac{2651\times 1483}{252\times 315}}{\displaystyle \frac{\sqrt[315]{x^{184}}}{x^{11}}}$$

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.