6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Logarithmic Differentiation

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Logarithmic Differentiation

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Use logarithmic differentiation to find the derivative of the given function.
$y=\tan^{x}x$

$\left(\tan x\right)^{x}\cdot\ln\left(\tan x\right)\cdot\sec^2x$

$\left(\tan x\right)^{x-1}\cdot\sec^2x$

$\ln\left(\tan x\right)+\frac{x\sec^2x}{\tan x}$

$\left(\ln\left(\tan x\right)+\frac{x\sec^2x}{\tan x}\right)\cdot\tan^{x}x$

Verified step by step guidance

Start by taking the natural logarithm of both sides of the equation y = (tan x)^x. This gives us ln(y) = ln((tan x)^x).

Use the property of logarithms that allows you to bring the exponent down: ln(y) = x * ln(tan x).

Differentiate both sides with respect to x. On the left side, use implicit differentiation: d/dx[ln(y)] = (1/y) * dy/dx. On the right side, apply the product rule to differentiate x * ln(tan x).

The product rule states that d/dx[u*v] = u'v + uv'. Here, u = x and v = ln(tan x). Differentiate u and v separately: u' = 1 and v' = (1/tan x) * sec^2(x) using the chain rule.

Substitute the derivatives back into the product rule: dy/dx = y * (ln(tan x) + x * (sec^2(x)/tan x)). Finally, replace y with (tan x)^x to express dy/dx in terms of x.