differentiate. y = ln x / x^13 dy / dx = □ (use parentheses to clearly denote the argument of each function.)
differentiate. y = ln x / x^13 dy / dx = □ (use parentheses to clearly denote the argument of each function.)
Answer
Explanation:
Step1: Apply quotient - rule
The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = \ln x$ and $v=x^{13}$.
Step2: Find $u^\prime$ and $v^\prime$
The derivative of $u=\ln x$ is $u^\prime=\frac{1}{x}$, and the derivative of $v = x^{13}$ using the power - rule $(x^n)^\prime=nx^{n - 1}$ is $v^\prime = 13x^{12}$.
Step3: Substitute into quotient - rule
$y^\prime=\frac{\frac{1}{x}\cdot x^{13}-\ln x\cdot13x^{12}}{(x^{13})^{2}}$.
Step4: Simplify the numerator and denominator
Simplify the numerator: $\frac{1}{x}\cdot x^{13}=x^{12}$ and the denominator is $x^{26}$. So $y^\prime=\frac{x^{12}-13x^{12}\ln x}{x^{26}}$. Factor out $x^{12}$ from the numerator: $y^\prime=\frac{x^{12}(1 - 13\ln x)}{x^{26}}$. Then cancel out $x^{12}$: $y^\prime=\frac{1 - 13\ln x}{x^{14}}$.
Answer:
$\frac{1 - 13\ln(x)}{x^{14}}$