Question

$\textsf{Find derivative of}\:\:\frac{(2x-7)^{2}}{x^{4}}$

Answer 1

$\boxed{\underline{\textsf{Question :}}}$

$\textsf{Find derivative of}\:\:\frac{(2x-7)^{2}}{x^{4}}\:\:\textsf{w. r. to x.}$

$\boxed{\underline{\textsf{Formulas :}}}$

$1.\:\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$

$2.\:\frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$

$3.\:\frac{d}{dx}(x^{n})=n\:x^{n-1}$

$\boxed{\underline{\textsf{Solution :}}}$

$\boxed{\boxed{\textsf{Method 1 :}}\:\boxed{\textsf{Using multiplication}}}$

$\textsf{Let us take}\:p=\frac{(2x-7)^{2}}{x^{4}}$

$\implies p=(2x-7)^{2}x^{-4}$

$\textsf{Differentiating w. r. to x, we get}$

$\frac{dp}{dx}=\frac{d}{dx}\{(2x-7)^{2}x^{-4}$

$=(2x-7)^{2}\frac{d}{dx}(x^{-4})$

$+x^{-4}\frac{d}{dx}(2x-7)^{2}$

$=\{(2x-7)^{2}(-4)x^{-4-1}\}$

$+\{2x^{-4}(2x-7)^{2-1}\frac{d}{dx}(2x-7)\}$

$=-4(2x-7)^{2}x^{-5}+4(2x-7)x^{-4}$

$=4(2x-7)x^{-4}\{-(2x-7)x^{-1}+1\}$

$=4(2x-7)x^{-4}(\frac{-2x+7+x}{x})$

$=\frac{4(2x-7)(7-x)x^{-4}}{x}$

$=\frac{4(2x-7)(7-x)}{x^{5}}$

$\to \boxed{\small{\frac{d}{dx}\frac{(2x-7)^{2}}{x^{4}}=\frac{4(2x-7)(7-x)}{x^{5}}}}$

$\boxed{\boxed{\textsf{Method 2 :}}\:\boxed{\textsf{Using division}}}$

$\textsf{Let us take}\:q=\frac{(2x-7)^{2}}{x^{4}}$

$\textsf{Differentiating w. r. to x, we get}$

$\frac{dq}{dx}=\frac{d}{dx}\frac{(2x-7)^{2}}{x^{4}}$

$=\frac{x^{4}\frac{d}{dx}(2x-7)^{2}-(2x-7)^{2}\frac{d}{dx}(x^{4})}{(x^{4})^{2}}$

$=\frac{\{2*2*x^{4}*(2x-7)\}-\{(2x-7)^{2}*4*x^{3}\}}{x^{8}}$

$=\frac{4(2x-7)x^{4}-4(2x-7)^{2}x^{3}}{x^{8}}$

$=\frac{4(2x-7)x^{3}\{x-(2x-7)\}}{x^{8}}$

$=\frac{4(2x-7)(7-x)}{x^{5}}$

$\to \boxed{\small{\frac{d}{dx}\frac{(2x-7)^{2}}{x^{4}}=\frac{4(2x-7)(7-x)}{x^{5}}}}$

$\boxed{\boxed{\textsf{Method 3 :}}\:\boxed{\textsf{Normal calculations}}}$

$\textsf{Now,}\:\:\frac{d}{dx} \frac{(2x-7)^{2}}{x^{4}}$

$=\frac{d}{dx}\frac{4x^{2}-28x+49}{x^{4}}$

$=\frac{d}{dx}(4x^{-2}-28x^{-3}+49x^{-4})$

$=\small{4\frac{d}{dx}(x^{-2})-28\frac{d}{dx}(x^{-3})+49\frac{d}{dx}(x^{-4})}$

$=-8x^{-3}+81x^{-4}-196x^{-5}$

$\to \boxed{\tiny{\bold{\frac{d}{dx} \frac{(2x-7)^{2}}{x^{4}}=-8x^{-3}+81x^{-4}-196x^{-5}}}}$

Answer 2

Answer:

Step-by-step explanation: