Name: Differentiation of Exponential Functions
Uploaded: 2019-07-19
Duration: 12 min 3 s
Description: To know how to find the derivative of exponential functions or multiple functions containing the exponential term. Let's watch this video and explore!

Question

Differentiate the following function with respect to $x$ : $\log_{e} (x^{x} + \csc^{2} (x))$ .

Accepted Answer

Let $z = \log_{e} (x^{x} + \csc^{2} (x))$

Differentiating both sides with respect to $x$ ,

$\frac{d z}{d x} = \frac{d}{d x} \{\log_{e} (x^{x} + \csc^{2} (x))\}$

$\frac{d z}{d x} = \frac{1}{(x^{x} + \csc^{2} (x))} \frac{d}{d x} (x^{x} + \csc^{2} (x))$

$\frac{d z}{d x} = \frac{1}{(x^{x} + \csc^{2} (x))} [\frac{d}{d x} (x^{x}) + \frac{d}{d x} \{\csc^{2} (x)\}] \dots (i)$

Let us assume $y = x^{x}$

Taking logarithm on both the sides,

$\log_{e} (y) = \log_{e} (x^{x})$

$\log_{e} (y) = x \log_{e} (x)$

Differentiating both the sides with respect to $x$ ,

$\frac{d}{d x} \{\log_{e} (y)\} = \frac{d}{d x} \{x \log_{e} (x)\}$

$\frac{1}{y} \frac{d y}{d x} = x \frac{d}{d x} \{\log_{e} (x)\} + \log_{e} (x) \frac{d}{d x} (x)$

$\frac{1}{y} \frac{d y}{d x} = x \frac{1}{x} + \log_{e} (x)$

$\Rightarrow \frac{d y}{d x} = y \{1 + \log_{e} (x)\}$

$\frac{d}{d x} (x^{x}) = x^{x} \{1 + \log_{e} (x)\}$

Putting this value in equation (i),

$\frac{d z}{d x} = \frac{1}{(x^{x} + \csc^{2} (x))} [x^{x} \{1 + \log_{e} (x)\} + \frac{d}{d x} \{\csc^{2} (x)\}]$

$\frac{d z}{d x} = \frac{1}{(x^{x} + \csc^{2} (x))} [x^{x} \{1 + \log_{e} (x)\} + \{2 \csc (x) (- \csc (x) \cot (x))\}]$

$\frac{d z}{d x} = \frac{1}{(x^{x} + \csc^{2} (x))} [x^{x} \{1 + \log_{e} (x)\} - \{2 \csc^{2} (x) \cot (x)\}]$

Differentiate the following function with respect to $x$ : $\log_{e} (x^{x} + \csc^{2} (x))$ .

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate the following function with respect to x: logexx+csc2x.

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate the following function with respect to $x$ : $\log_{e} (x^{x} + \csc^{2} (x))$ .