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Chapter 2: Q 20. (page 222)

Find the derivatives of the functions: $f (x) = e^{3 x} \ln (x^{2} + 1)$

Short Answer

Expert verified

The required answer is $\frac{2 x e^{3 x}}{x^{2} + 1} + 3 e^{3 x} \ln (x^{2} + 1)$ .

Step by step solution

01

Step 1. Given information.

The given function is: $f (x) = e^{3 x} \ln (x^{2} + 1)$

02

Step 2. Find the derivative of the given function.

$f (x) = e^{3 x} \ln (x^{2} + 1) f' (x) = \frac{d}{d x} (e^{3 x} \ln (x^{2} + 1)) = e^{3 x} \frac{d}{d x} \ln (x^{2} + 1) + \ln (x^{2} + 1) \frac{d}{d x} e^{3 x} = e^{3 x} \frac{1}{x^{2} + 1} 脳 2 x + \ln (x^{2} + 1) 3 e^{3 x} = \frac{2 x e^{3 x}}{x^{2} + 1} + 3 e^{3 x} \ln (x^{2} + 1)$

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Most popular questions from this chapter

Use the definition of the derivative to prove the following special case of the product rule

$\frac{d}{d x} (x^{2} f (x)) = 2 x f (x) + x^{2} f' (x)$

The following reciprocal rules tells us hoe to differentiate the reciprocal of a function

$\frac{d}{d x} (\frac{1}{f (x)}) = - \frac{1}{{[f (x)]}^{2}}$

Prove this using

a) definition of the derivative

b) by using the quotient rule

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = \frac{3 x + 1}{\sqrt{x^{2} + 1}}$

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = x^{- \frac{1}{2}} {(x^{2} - 1)}^{3}$

Prove the difference rule in two ways

a) using definition of the derivative

b) using sum and constant multiple rules

See all solutions

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