Question

Given f(x) = (x^2)/(3 + 8x), find f prime (x) and f double prime (x).

Answer 1

The derivative of a function f(x) is given by

ddxf(x)=g(x)

Then the integration of g(x) is given by

∫g(x)⋅dx=f(x)+C

We can differentiate power functions , so we can also integrate power functions.

Power Rule :

∫xndx=xn+1n+1+C

Answer and Explanation:

f″(x)=20x3+12x2+8

Integrating using Power rule -

f′(x)=20x44+12x33+8x+C1f′(x)=5x4+4x3+8x+C1

Integrating using power rule again gives -

f(x)=5x55+4x44+8x22+C1x+C2f(x)=x5+x4+4x2+C1x+C2

It is given that f(0)=6 and f(1)=7

Replacing x=0 in f(x)

f(0)=C2=6 

Replacing x=1 in f(x)

f(1)=1+1+4+C1+C2=6+C1+6=C1+12

C1+12=7 

C1=−5

Using C1=−5 and C2=6 the function {eq} f(x) {\eq} is represented by -

f(x)=x5+x4+4x2−5x+6 

Answer 2

$\huge{\boxed{ \red {\bf{\ulcorner Answer \: \lrcorner}}}}$


$<b>$
{eq}f'(x) = \frac{2x(3+8x) - x^2(8)}{(3+8x)^2} \\ f'(x) = \frac{6x+16x^2 - 8x^2}{(3+8x)^2} \\ f'(x)



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