Question

for y = x^6 - 7x^3 + 6x, find $\frac{d^{5}y}{dx^{5}}$. $\frac{d^{5}y}{dx^{5}}=square$

Explanation:

Step1: Recall power - rule for differentiation

The power - rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$.

Step2: Differentiate $y = x^6-7x^3 + 6x$ once

$\frac{dy}{dx}=\frac{d}{dx}(x^6)-7\frac{d}{dx}(x^3)+6\frac{d}{dx}(x)=6x^{5}-21x^{2}+6$.

Step3: Differentiate $\frac{dy}{dx}$ a second time

$\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}(6x^{5})-21\frac{d}{dx}(x^{2})+\frac{d}{dx}(6)=30x^{4}-42x$.

Step4: Differentiate $\frac{d^{2}y}{dx^{2}}$ a third time

$\frac{d^{3}y}{dx^{3}}=\frac{d}{dx}(30x^{4})-42\frac{d}{dx}(x)=120x^{3}-42$.

Step5: Differentiate $\frac{d^{3}y}{dx^{3}}$ a fourth time

$\frac{d^{4}y}{dx^{4}}=\frac{d}{dx}(120x^{3})-\frac{d}{dx}(42)=360x^{2}$.

Step6: Differentiate $\frac{d^{4}y}{dx^{4}}$ a fifth time

$\frac{d^{5}y}{dx^{5}}=\frac{d}{dx}(360x^{2}) = 720x$.

Answer:

$720x$