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Chapter 2: Q. 49 (page 233)

Find the derivatives of each of the functions in Exercises 17鈥�50. In some cases it may be convenient to do some preliminary algebra
$f (x) = \sec (1 + \tan^{- 1} x)$

Short Answer

Expert verified

The derivative is $\frac{\sec (1 + \tan^{- 1} x) \tan (1 + \tan^{- 1} x)}{1 + x^{2}}$ .

Step by step solution

01

Step 1. Given Information.

The given function is $f (x) = \sec (1 + \tan^{- 1} x)$

02

Step 2. Preliminary Algebra.

We know,

$\frac{d}{d x} (\tan^{- 1} x) = \frac{1}{1 + x^{2}} C h a i n R u l e : (f 鈭� u)^{'} (x) = f^{'} (u (x)) u^{'} (x)$

03

Step 3. Derivative of the function.

The derivative of the function is,

$\frac{d}{d x} (\sec (1 + \tan^{- 1} x)) = \sec (1 + \tan^{- 1} x) \tan (1 + \tan^{- 1} x) \frac{d}{d x} (1 + \tan^{- 1} x) = \sec (1 + \tan^{- 1} x) \tan (1 + \tan^{- 1} x) (\frac{d}{d x} 1 + \frac{d}{d x} \tan^{- 1} x) = \sec (1 + \tan^{- 1} x) \tan (1 + \tan^{- 1} x) (\frac{1}{1 + x^{2}}) = \frac{\sec (1 + \tan^{- 1} x) \tan (1 + \tan^{- 1} x)}{1 + x^{2}}$

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Most popular questions from this chapter

Prove the difference rule in two ways

a) using definition of the derivative

b) using sum and constant multiple rules

For each function $f$ graphed in Exercises 65-68, determine the values of $x$ at which $f$ fails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

For each function $f$ graphed in Exercises 65-68, determine the values of $x$ at which $f$ fails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

Prove that if f is any cubic polynomial function $f (x) = a x^{3} + b x^{2} + c x + d$ then the coefficients of f are completely determined by the values of f(x) and its derivative at x=0 as follows

$a = \frac{f "' (0)}{6}; b = \frac{f " (0)}{2}; c = f' (0); d = f (0)$

Use the differentiation rules developed in this section to find

the derivatives of the functions

$f (x) = 4 - x^{7}$

See all solutions

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