find the derivative of the following function. f(x)=10x^3 - 31x + e^4 f(x)=□

Explanation:

Step1: Apply power - rule to $10x^3$

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $y = 10x^3$, $a = 10$ and $n = 3$. So the derivative is $10\times3x^{3 - 1}=30x^2$.

Step2: Apply power - rule to $-31x$

For $y=-31x$, $a=-31$ and $n = 1$. Using the power - rule $y^\prime=-31\times1x^{1 - 1}=-31$.

Step3: Derivative of a constant

Since $e^4$ is a constant, its derivative is 0.

Step4: Sum of derivatives

The derivative of $f(x)=10x^3-31x + e^4$ is the sum of the derivatives of its terms. So $f^\prime(x)=30x^2-31+0$.

Answer:

$30x^2 - 31$