Question

differentiate the function.
g(x)=\frac{5}{4}x^{2}-4x + 17
g(x)=

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differentiate the function.
f(x)=x^{3}(x + 4)
f(x)=

Explanation:

Step1: Recall power - rule of differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$, and the derivative of a sum/difference of functions is the sum/difference of their derivatives. For $g(x)=\frac{5}{4}x^{2}-4x + 17$.

Step2: Differentiate each term

For the first term $\frac{5}{4}x^{2}$, using the power - rule with $a=\frac{5}{4}$ and $n = 2$, we get $\frac{d}{dx}(\frac{5}{4}x^{2})=\frac{5}{4}\times2x=\frac{5}{2}x$. For the second term $-4x$, with $a=-4$ and $n = 1$, we get $\frac{d}{dx}(-4x)=-4$. For the third term $17$ (a constant), $\frac{d}{dx}(17)=0$.

Step3: Combine the derivatives

$g^\prime(x)=\frac{5}{2}x-4$.

For $f(x)=x^{3}(x + 4)=x^{4}+4x^{3}$.

Step4: Differentiate $f(x)$ term - by - term

For the first term $x^{4}$, using the power - rule with $a = 1$ and $n = 4$, we get $\frac{d}{dx}(x^{4})=4x^{3}$. For the second term $4x^{3}$, with $a = 4$ and $n = 3$, we get $\frac{d}{dx}(4x^{3})=4\times3x^{2}=12x^{2}$.

Step5: Combine the derivatives of $f(x)$

$f^\prime(x)=4x^{3}+12x^{2}$.

Answer:

$g^\prime(x)=\frac{5}{2}x - 4$
$f^\prime(x)=4x^{3}+12x^{2}$