Answer

Verified

445.2k+ views

**Hint:**Differentiation- It is the action of computing a derivative.

The derivative of a function \[y = f\left( x \right)\] of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x.

It is denoted by \[dy/dx.\]

Some formulae of finding differentiation

$\dfrac{d}{{dx}}(ax) = a$

$\dfrac{d}{{dx}}(x) = 1$

$\dfrac{d}{{dx}}(c) = 0$ \[\left[ {c = constant} \right]\]

$\dfrac{d}{{dx}}({x^n}) = {x^{n - 1}}$

$\dfrac{d}{{dx}}({e^n}) = {e^x}$

$\dfrac{d}{{dx}}(ax \pm b) = \dfrac{d}{{dx}}(ax) \pm \dfrac{d}{{dx}}(b)$

$\dfrac{d}{{dx}}(\log x) = \dfrac{1}{x}$

$\dfrac{d}{{dx}}(\sin x) = \cos x$

$\dfrac{d}{{dx}}(cosx) = - \sin x$

$\dfrac{d}{{dx}}(\tan x) = \sec {x^2}$

$\dfrac{d}{{dx}}(\sec x) = \sec x\tan x$

$\dfrac{d}{{dx}}(\cot x) = - \cos e{c^2}x$

$\dfrac{d}{{dx}}(\cos ecx) = \cos ecx\cot x$

**Complete step by step solution:**

$y = {x^x}{e^{2x + 5}}$

Taking log both the sides

$\log y = \log ({x^x}{e^{2x + 5}})$

or

$\log y = \log {x^x} + \log {e^{2x + 5}}$

Differentiating both the sides

$\dfrac{d}{{dx}}(\log y) = \dfrac{d}{{dx}}(\log {x^x}) + \dfrac{d}{{dx}}[\log {e^{2x + 5}}]$

$\dfrac{1}{y}\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(x\log x) + \dfrac{d}{{dx}}[(2x + 5){\log _e}e]$

$\dfrac{1}{y}\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(x) \times \log x + x\dfrac{d}{{dx}}(\log x) + \dfrac{d}{{dx}}(2x + 5)$

$\dfrac{1}{y}\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(x) \times \log x + x\dfrac{d}{{dx}}(\log x) + \dfrac{d}{{dx}}(2x + 5)$

$\dfrac{1}{y}\dfrac{{dy}}{{dx}} = (1)\log x + x \times \left( {\dfrac{1}{x}} \right) + 2$

$\dfrac{1}{y}\dfrac{{dy}}{{dx}} = \log x + 1 + 2$

$\dfrac{{dy}}{{dx}} = (\log x + 3) \times y$

or $\dfrac{{dy}}{{dx}} = [\log x + 3][{x^x}{e^{2x + 5}}]$

So derivative of $y = {e^x}{e^{2x + 5}}$ is $[\log x + 3]({x^x}{e^{2x + 5}})$.

**Note:**1.Students usually forget to use u.v formula i.e. differentiation by parts for finding derivation of two functions which are in multiplication with each other.

U.v formula \[\dfrac{{d(uv)}}{{dx}} = \dfrac{{v.d\left( u \right)}}{{dx}} + \dfrac{{u.d\left( v \right)}}{{dx}}\]

(differentiation by parts)

One function will remain constant and the 2nd function to be differentiated.

Then the 2nd function will remain constant and the 1st one is to be differentiated.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

At which age domestication of animals started A Neolithic class 11 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE