Question

1. find the derivative of y = 5^x + 2/x^6. answer =

Explanation:

Step1: Recall derivative rules

The derivative of $a^x$ is $a^x\ln a$ and the derivative of $x^n$ is $nx^{n - 1}$.

Step2: Differentiate $5^x$

The derivative of $y_1 = 5^x$ is $y_1'=5^x\ln 5$ according to the rule for $a^x$ derivative.

Step3: Rewrite and differentiate $\frac{2}{x^6}$

Rewrite $\frac{2}{x^6}$ as $2x^{-6}$. Using the power - rule, its derivative $y_2'=2\times(- 6)x^{-6 - 1}=-12x^{-7}=-\frac{12}{x^7}$.

Step4: Use sum - rule of derivatives

Since $y = 5^x+\frac{2}{x^6}=y_1 + y_2$, by the sum - rule of derivatives $y'=y_1'+y_2'$. So $y'=5^x\ln 5-\frac{12}{x^7}$.

Answer:

$5^x\ln 5-\frac{12}{x^7}$