Question

for the function ( y = 2x^5 ), find ( \frac{d^4 y}{dx^4} ). ( \frac{d^4 y}{dx^4} = square )

Explanation:

Step1: Find the first derivative

Using the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, for $y = 2x^5$, the first derivative $y'=\frac{dy}{dx}=2\times5x^{5 - 1}=10x^4$.

Step2: Find the second derivative

Differentiate $y' = 10x^4$ again. Using the power rule, $y''=\frac{d^2y}{dx^2}=10\times4x^{4 - 1}=40x^3$.

Step3: Find the third derivative

Differentiate $y'' = 40x^3$. Using the power rule, $y'''=\frac{d^3y}{dx^3}=40\times3x^{3 - 1}=120x^2$.

Step4: Find the fourth derivative

Differentiate $y''' = 120x^2$. Using the power rule, $\frac{d^4y}{dx^4}=120\times2x^{2 - 1}=240x$.

Answer:

$240x$