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)

= sec² 0.14112 × (-0.989992)

= - sec² 0.14112 × 0.989992

= -(1/cos 0.14112)² × 0.989992

By using a calculator.

f^'(-3) = -1.009

Problem 6 :

f(x) = 2^In(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