find the derivative of ( f(x) ).\n\n( f(x)=6^{x}+e^{2 x} )\n\n( f^{prime}(x)= )

Answer:

$6^{x}\ln 6 + 2e^{2x}$

Explanation:

Step1: Differentiate $6^{x}$

Use the formula $\frac{d}{dx}a^{x}=a^{x}\ln a$. For $a = 6$, we have $\frac{d}{dx}6^{x}=6^{x}\ln 6$.

Step2: Differentiate $e^{2x}$

Use the chain - rule. Let $u = 2x$, then $\frac{d}{dx}e^{u}=e^{u}\cdot\frac{du}{dx}$. Since $\frac{du}{dx}=2$, we have $\frac{d}{dx}e^{2x}=e^{2x}\cdot2 = 2e^{2x}$.

Step3: Sum the derivatives

By the sum rule of differentiation $\frac{d}{dx}(u + v)=\frac{du}{dx}+\frac{dv}{dx}$. Here $u = 6^{x}$ and $v=e^{2x}$, so $f^{\prime}(x)=\frac{d}{dx}(6^{x})+\frac{d}{dx}(e^{2x})=6^{x}\ln 6+2e^{2x}$.