ESTIMATE THE DERIVATIVE OF A FUNCTION AT A GIVEN POINT

Estimate the derivative at the given point by using a calculator.

Problem 1 :

f(x) = x√(2 - x); find f'(-10).

Solution :

f(x) = x√(2 - x)

Using product rule.

d/dx[f ⋅ g] = f'g + fg'

By using a calculator.

f'(-10) = 4.907

Problem 2 :

f(x) = sec(5x); find f'(2).

Solution :

f(x) = sec(5x)

Using product rule.

d/dx[f ⋅ g] = f'g + fg'

f'(x) = sec (5x)tan(5x) ⋅ 5

f'(x) = 5 sec 5x tan 5x

To find f'(2) :

f'(2) = 5 sec 5 ⋅ 2 tan 5 ⋅ 2

= 5 sec 10 tan 10

= 5 ⋅ (1/cos 10) tan 10

By using a calculator.

f'(2) = -3.863

Problem 3 :

f(x) = In(√x); find f'(1).

Solution :

f(x) = In(√x)

Problem 4 :

Solution :

Problem 5 :

f(x) = tan(sin x); find f'(-3)

Solution :

Given, f(x) = tan(sin x)

Using product rule.

d/dx[f ⋅ g] = f'g + fg'

f'(x) = sec2(sin x) cos x

f'(-3) = sec2(sin (-3)) cos (-3)

f'(-3) = sec2(sin 3) (cos 3)

= (1 + tan2(sin 3)) (cos 3)

= (1 + tan2 0.14112) × (-0.989992)

= sec0.14112 × (-0.989992)

= - sec0.14112 × 0.989992

= -(1/cos 0.14112)2 × 0.989992

By using a calculator.

f'(-3) = -1.009

Problem 6 :

f(x) = 2In(x); find f'(2).

Solution :

Problem 7 :

The model f(t) = x/cos x measures the height of bird in meters where t is seconds. Find f'(2).

Solution :

f'(2) = 8.098 m/sec

Problem 8 :

The model f(t) = sin2(t) measures the depth of a submarine measured in feet where t is minutes. Find f'(12.5).

Solution :

Given, f(t) = sin2(t)

f'(t) = 2 (sin t) (cos t)

To find f'(12.5) :

f'(12.5) = 2 (sin 12.5) (cos 12.5)

= 2 (-0.066) (0.9978)

= -0.132 × 0.9978

f'(12.5)  = -0.13 ft/min

Problem 9 :

The model f(t) = √x - 1/(x - 1) measures the number of stocks sold where t is days. Find f'(12).

Solution :

f'(12)  = 0.1526 stocks/day

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