FIND THE DERIVATIVE OF EXPONENTIAL FUNCTIONS

To find the derivative of exponential functions, we use the rule given below.

d(ex)/dx = ex

d(eax)/dx = aex

Find the derivative function:

Problem 1 :

y = xe-2x

Solution :

y = xe-2x

Here two differentiable functions are multiplied, so we use product rule to find the derivative.

u = x and v = e-2x

u' = 1 and v' = -2e-2x

d(uv) = uv' + vu'

= x(-2e-2x) + e-2x (1)

dy/dx = -2xe-2x + e-2x

Problem 2 :

y = x³e-x

Solution :

Here two differentiable functions are multiplied, so we use product rule to find the derivative.

u =  and v = e-x

u' = 3x2 and v' = -e-x

d(uv) = uv' + vu'

x³(-e-x) + 3x2(e-x)

dy/dx = 3x² e-x - x³ e-x

Problem 3 :

y = x³ - xe4x

Solution :

= d/dx [x³ - xe4x]

u = x and v = e-4x

u' = 1 and v' = -4e-x

= 3x² - [x(-4e-x) + e-4x (1)]

dy/dx = 3x² - e4x - 4xe4x

Problem 4 :

y = (x² - 6) e8x

Solution :

= d/dx [(x² - 6) e8x]

u = x² - 6 and v = e8x

u' = 2x and v' = 8e8x

= (x² - 6) 8e8x + 2xe8x

Factoring e8x, we get

dy/dx = e8x (8x² - 48 + 2x)

dy/dx = 2e8x (4x² + x - 24)

Problem 5 :

y = √x ex

Solution :

= d/dx [√x ex]

= d/dx [√x] ex + √x d/dx [ex]

= 1/2 x1/2-1 ex + √x ex

dy/dx = √x ex + ex/2√x

Problem 6 :

y = 4e2x^2

Solution :

= d/dx [4e2x^2]

= 4 · d/dx [e2x^2]

= 4e2x^2 · d/dx [2x²]

= 4e2x^2 · (4x)

dy/dx = 16xe2x^2

Problem 7 :

y = xex^2

Solution :

= d/dx [xex^2]

u = x and v = ex^2

u' = 1 and v' = ex^2 (2x)

= x (2xex^2) + ex^2 (1)

= ex^2 + xex^2 (2x)

dy/dx = ex^2 + 2x2ex^2

Problem 8 :

y = e(e^x)

Solution :

= d/dx [e(e^x)]

= e(e^x) · d/dx [ex]

dy/dx = e(e^x) · ex

Problem 9 :

y = e2x+1 / 2x + 7

Solution :

= d/dx [e2x+1 / 2x + 7]

Problem 10 :

y = e3x/x2

Solution :

= d/dx [e3x/x2]

u = e3x and u' = 3e3x

v = x2 and v' = 2x

= [x2(3e3x) - e3x (2x)] / x4

e3x (3x2 - 2x) / x4

= e3x (3x - 2) / x3

Problem 11 :

y = e√x

Solution :

= d/dx [e√x]

= e√x · (1/2 x1/2-1)

dy/dx = e√x / 2√x

Problem 12 :

y = ex + 1 / ex - 1

Solution :

Recent Articles

1. Finding Range of Values Inequality Problems

May 21, 24 08:51 PM

Finding Range of Values Inequality Problems

2. Solving Two Step Inequality Word Problems

May 21, 24 08:51 AM

Solving Two Step Inequality Word Problems