differentiate the following function. f(x)=x^8e^9x f(x)=

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = x^{8}$ and $v = e^{9x}$.

Step2: Differentiate $u$

Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, for $u = x^{8}$, we have $u^\prime=\frac{d}{dx}(x^{8}) = 8x^{7}$.

Step3: Differentiate $v$

Using the chain - rule $\frac{d}{dx}(e^{ax})=ae^{ax}$, for $v = e^{9x}$, we have $v^\prime=\frac{d}{dx}(e^{9x})=9e^{9x}$.

Step4: Calculate $f^\prime(x)$

By the product - rule $f^\prime(x)=u^\prime v+uv^\prime$. Substitute $u = x^{8}$, $u^\prime = 8x^{7}$, $v = e^{9x}$, and $v^\prime = 9e^{9x}$ into the formula: $f^\prime(x)=8x^{7}e^{9x}+x^{8}\cdot9e^{9x}=x^{7}e^{9x}(8 + 9x)$.

Answer:

$x^{7}e^{9x}(8 + 9x)$