Question

for the function y = 6x^7+3xe^{2x - 6} find \frac{dy}{dx}. factor your answer where possible. answer: \frac{dy}{dx}=

Explanation:

Step1: Differentiate term - by - term

The derivative of a sum of functions is the sum of the derivatives of the functions. So, $\frac{dy}{dx}=\frac{d}{dx}(6x^{7})+\frac{d}{dx}(3xe^{2x - 6})$.

Step2: Differentiate $6x^{7}$

Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, for $a = 6$ and $n = 7$, we have $\frac{d}{dx}(6x^{7})=6\times7x^{7 - 1}=42x^{6}$.

Step3: Differentiate $3xe^{2x - 6}$ using the product rule

The product rule states that if $u(x)$ and $v(x)$ are functions, then $\frac{d}{dx}(u(x)v(x))=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)$. Let $u = 3x$ and $v = e^{2x - 6}$. Then $u^{\prime}=3$ and $v^{\prime}=2e^{2x - 6}$ (by the chain - rule, since the derivative of $e^{f(x)}$ is $f^{\prime}(x)e^{f(x)}$ and $f(x)=2x - 6$ with $f^{\prime}(x)=2$). So, $\frac{d}{dx}(3xe^{2x - 6})=3e^{2x - 6}+3x\times2e^{2x - 6}=3e^{2x - 6}(1 + 2x)$.

Step4: Combine the results

$\frac{dy}{dx}=42x^{6}+3e^{2x - 6}(1 + 2x)$.

Answer:

$\frac{dy}{dx}=42x^{6}+3e^{2x - 6}(2x + 1)$