for the function y = 4x^5, find d^4y/dx^4. d^4y/dx^4 =

Explanation:

Step1: Recall power - rule for differentiation

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

Step2: First - order derivative

For $y = 4x^{5}$, using the power - rule, $\frac{dy}{dx}=4\times5x^{5 - 1}=20x^{4}$.

Step3: Second - order derivative

Differentiate $\frac{dy}{dx}=20x^{4}$ again. $\frac{d^{2}y}{dx^{2}}=20\times4x^{4 - 1}=80x^{3}$.

Step4: Third - order derivative

Differentiate $\frac{d^{2}y}{dx^{2}}=80x^{3}$ again. $\frac{d^{3}y}{dx^{3}}=80\times3x^{3 - 1}=240x^{2}$.

Step5: Fourth - order derivative

Differentiate $\frac{d^{3}y}{dx^{3}}=240x^{2}$ again. $\frac{d^{4}y}{dx^{4}}=240\times2x^{2 - 1}=480x$.

Answer:

$480x$