Question

oblicz pochodną f(x)=(x^2 * sinx)^2

Answer 1

f(x) = ( x^2 * sin x )^2

czyli f(x) = ( x^2 *sin x) *( x^2 * sin x)

Stosujemy wzór na pochodną iloczynu funkcji

 

f'(x) = [ x^2 *sin x ]' [x^2 *sin x] + [ x^2*sin x]*[x^2 *sin x}'

Obliczmy

[ x^2 *sin x]' = 2x*sin x + x^2 * cos x

zatem

f'(x) = [ 2x *sin x + x^2*cos x]*[x^2 *sin x] +

+ [x^2 *sin x]*[2x *sin x + x^2 *cos x] =

= 2 *[2 x^3 *sin ^2 x + x^4 * sin x *cos x] =

= 2 x^4 *sin x * cos x + 4 x^3 * sin^2 x

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sin^2 x   <-- sinus kwadrat x

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II sposób:

f(x) = ( x^2 *sin x)^2 = x^4 * sin^2 x

zatem

f' (x) =  4 *x^3 * sin^2 x + x^4 * 2*sin x * cos x =

= 2 *x^4 *sin x *cos x + 4*x^3* sin^2 x

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Answer 2