Question

find $\frac{dy}{dx}$ for the indicated function y.
y = 10^x + e^2
$\frac{dy}{dx}=square$ (type an exact answer.)

Explanation:

Step1: Recall derivative rules

The derivative of a sum of functions is the sum of their derivatives, i.e., if $y = u + v$, then $\frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}$. Also, the derivative of $a^x$ with respect to $x$ is $a^x\ln a$ and the derivative of a constant is 0.

Step2: Differentiate $10^x$

For $u = 10^x$, by the formula $\frac{d(a^x)}{dx}=a^x\ln a$, we have $\frac{du}{dx}=10^x\ln 10$.

Step3: Differentiate $e^2$

Since $v = e^2$ is a constant (because $e^2\approx 7.39$ is a fixed - value number), $\frac{dv}{dx}=0$.

Step4: Find $\frac{dy}{dx}$

Using $\frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}$, we get $\frac{dy}{dx}=10^x\ln 10+0$.

Answer:

$10^x\ln 10$