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 functions w.r.t. $x$ : $x^{x^{x}}$

Accepted Answer

Let $y = x^{x^{x}}$ ..... $(i)$

Taking logarithm both the sides,

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

$\log_{e} (y) = x^{x} \log_{e} (x)$ ..... $(ii)$ $\{∵ \log_{e} (m^{n}) = n \log_{e} (m)\}$

Again taking logarithm both the sides,

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

$\log_{e} (\log_{e} (y)) = \log_{e} (x^{x}) + \log_{e} (\log_{e} (x))$ $\{∵ \log_{e} (m n) = \log_{e} (m) + \log_{e} (n)\}$

$\log_{e} (\log_{e} (y)) = x \log_{e} (x) + \log_{e} (\log_{e} (x))$ $\{∵ \log_{e} (m^{n}) = n \log_{e} (m)\}$

Differentiating both the sides with respect to $x$ ,

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

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

As we know by product rule of differentiation $\frac{d}{d x} [f (x) g (x)] = f (x) g' (x) + f' (x) g (x)$ ,

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

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

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

By equation $(i)$ and $(ii)$ ,

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

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

Differentiate the following functions w.r.t. $x$ : $x^{x^{x}}$

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate the following functions w.r.t. x: xxx

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate the following functions w.r.t. $x$ : $x^{x^{x}}$