Question

find the first and second derivatives of the function. f(x)=x^5 - x^4 + 2x^3 - 4x^2 + 2x - 5 f(x)= f(x)=

Explanation:

Step1: Apply power - rule for first derivative

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For $f(x)=x^{5}-x^{4}+2x^{3}-4x^{2}+2x - 5$, we have $f^\prime(x)=5x^{4}-4x^{3}+6x^{2}-8x + 2$.

Step2: Apply power - rule for second derivative

Differentiate $f^\prime(x)=5x^{4}-4x^{3}+6x^{2}-8x + 2$ using the power - rule again. So $f^{\prime\prime}(x)=20x^{3}-12x^{2}+12x - 8$.

Answer:

$f^\prime(x)=5x^{4}-4x^{3}+6x^{2}-8x + 2$; $f^{\prime\prime}(x)=20x^{3}-12x^{2}+12x - 8$