Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the derivative of $q(x)$ . $\newline$ $q(x) = \ln\left(\frac{x}{e}\right)$ $\newline$ $q'\left(x\right) =$ ______

View step-by-step help

View step-by-step help

Find derivatives using the quotient rule I

Full solution

Q. Find the derivative of

q(x)

.

\newline

q(x) = \ln\left(\frac{x}{e}\right)

\newline

q'\left(x\right) =

______

Simplifying q(x): Let's first simplify the function $q(x) = \ln(\frac{x}{e})$ . We can use the property of logarithms that $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$ to rewrite q(x). $\newline$ $q(x) = \ln(x) - \ln(e)$ $\newline$ Since $\ln(e)$ is a constant ( $\ln(e) = 1$ ), we can simplify further: $\newline$ $q(x) = \ln(x) - 1$
Finding the derivative of q(x): Now, we need to find the derivative of q(x) with respect to x. The derivative of $\ln(x)$ with respect to x is $\frac{1}{x}$ , and the derivative of a constant is $0$ . $\newline$ So, $q'(x) = \frac{d}{dx} [\ln(x) - 1]$ $\newline$ $q'(x) = \frac{d}{dx} [\ln(x)] - \frac{d}{dx} [1]$ $\newline$ $q'(x) = \frac{1}{x} - 0$
Final derivative of q(x): We can now write the final derivative of q(x): $\newline$ q'(x) = \frac{ $1$ }{x} $\newline$ This is the derivative of the function q(x) = \ln\left(\frac{x}{e}\right).

More problems from Find derivatives using the quotient rule I

Question

Find the derivative of

f(x)

\newline

f(x) = \cos(x)

\newline

f'(x) =

Posted 5 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \tan(x)

\newline

f'(x) =

Posted 6 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = e^x

\newline

f'\left(x\right) =

Posted 6 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \ln(x)

\newline

f'(x) =

Posted 6 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \tan^{-1}{x}

\newline

f'(x) =

Posted 6 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = x e^x

\newline

f'\left(x\right) =

Posted 6 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \frac{e^x}{x}

\newline

f'(x) =

Posted 6 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = e^{(x + 1)}

\newline

f'\left(x\right) =

Posted 5 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \sqrt{x+3}

\newline

f'(x) =

Posted 6 months ago

Question

Find the derivative of

g(x)

\newline

g(x) = \ln(2x)

\newline

g^{\prime}(x) =

Posted 7 months ago

Related topics

Algebra - Order of Operations Algebra - Distributive Property `X` and `Y` Axes Geometry - Scalene Triangle Common Multiple Geometry - Quadrant