find the derivative of $f(x)=x^{10}e^{7.5x}$.

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Here, $u = x^{10}$ and $v = e^{7.5x}$.

Step2: Differentiate $u = x^{10}$

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $u^\prime=\frac{d}{dx}(x^{10}) = 10x^{9}$.

Step3: Differentiate $v = e^{7.5x}$

Using the chain - rule. If $y = e^{u}$ and $u = 7.5x$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Since $\frac{d}{du}(e^{u}) = e^{u}$ and $\frac{d}{dx}(7.5x)=7.5$, we get $v^\prime = 7.5e^{7.5x}$.

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

Substitute $u$, $u^\prime$, $v$, and $v^\prime$ into the product - rule formula: $f^\prime(x)=u^\prime v+uv^\prime=10x^{9}e^{7.5x}+x^{10}\cdot7.5e^{7.5x}$.

Answer:

$10x^{9}e^{7.5x}+7.5x^{10}e^{7.5x}$