What is the diffquot ((f(x)(x+h(x))-f(x)))/(h(x),f(x))= 3/((x^2)) ?

The diffquot ((f(x)(x+h(x))-f(x)))/(h(x),f(x))= 3/((x^2)) is -(3(2x+Δ))/(x^2(x+Δ)^2)

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Difference Quotient (f(x+h)-f(x))/h , f(x)=3/(x^2)

difference quotient

\frac{( f ( x ) ( x + h ( x ) ) - f ( x ) )}{h ( x ) , f ( x )} = \frac{3}{( x ^{2} )}

- \frac{3 ( 2 x + Δ )}{x ^{2} ( x + Δ ) ^{2}}

Solution steps

The difference quotient of

\frac{( f ( x ) ( x + h ( x ) ) - f ( x ) )}{h ( x ) , f ( x )} (x)

is given by

\frac{\frac{( f ( x ) ( x + h ( x ) ) - f ( x ) )}{h ( x ) , f ( x )} ( x + Δ ) - \frac{( f ( x ) ( x + h ( x ) ) - f ( x ) )}{h ( x ) , f ( x )} ( x )}{Δ}

Substitute

x

with

x + Δ

\frac{( f ( x ) ( x + h ( x ) ) - f ( x ) )}{h ( x ) , f ( x )} (x + Δ) = \frac{3}{( ( x + Δ ) ^{2} )}

= \frac{\frac{3}{( ( x + Δ ) ^{2} )} - \frac{3}{( x ^{2} )}}{Δ}

Simplify

\frac{\frac{3}{( ( x + Δ ) ^{2} )} - \frac{3}{( x ^{2} )}}{Δ} : - \frac{3 ( 2 x + Δ )}{x ^{2} ( x + Δ ) ^{2}}

= - \frac{3 ( 2 x + Δ )}{x ^{2} ( x + Δ ) ^{2}}

- \frac{3 ( 2 x + Δ )}{x ^{2} ( x + Δ ) ^{2}}

- \frac{d}{d x} (sec (2 x))

- \frac{d}{d x} y = x e^{x}

- \frac{d}{d x} (\frac{x}{2})

X

- \frac{d}{d x} (e^{\frac{- x}{2}})

What is the diffquot ((f(x)(x+h(x))-f(x)))/(h(x),f(x))= 3/((x^2)) ?
The diffquot ((f(x)(x+h(x))-f(x)))/(h(x),f(x))= 3/((x^2)) is -(3(2x+Δ))/(x^2(x+Δ)^2)