Here, we show you a step-by-step solved example of derivative of logarithmic functions. This solution was automatically generated by our smart calculator:
Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$
Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$
The limit of a constant is just the constant
Rewrite the product inside the limit as a fraction
Plug in the value $0$ into the limit
$\ln(0)$ grows unbounded towards minus infinity
An expression divided by zero tends to infinity
If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{\ln\left(x\right)}{\frac{1}{x}}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form
We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately
Find the derivative of the numerator
The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$
Find the derivative of the denominator
Apply the quotient rule for differentiation, which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
The derivative of the constant function ($1$) is equal to zero
The derivative of the linear function is equal to $1$
$x+0=x$, where $x$ is any expression
Divide fractions $\frac{\frac{1}{x}}{\frac{-1}{x^2}}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
Simplify the fraction by $x$
Divide fractions $\frac{1}{\frac{-1}{x}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
After deriving both the numerator and denominator, the limit results in
The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$
Evaluate the limit $\lim_{x\to0}\left(x\right)$ by replacing all occurrences of $x$ by $0$
Evaluate the limit $\lim_{x\to0}\left(x\right)$ by replacing all occurrences of $x$ by $0$
Multiply $-1$ times $0$
Calculate the power $e^{0}$
Access detailed step by step solutions to thousands of problems, growing every day!
Most popular problems solved with this calculator: