FIND THE DERIVATIVE OF EXPONENTIAL FUNCTIONS

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 = x³ 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 - xe^4x

Solution :

= d/dx [x³ - xe^4x]

u = x and v = e^-4x

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

= 3x² - [x(-4e^-x) + e^-4x (1)]

dy/dx = 3x² - e^4x - 4xe^4x

Problem 4 :

y = (x² - 6) e^8x

Solution :

= d/dx [(x² - 6) e^8x]

u = x² - 6 and v = e^8x

u' = 2x and v' = 8e^8x

= (x² - 6) 8e^8x + 2xe^8x

Factoring e^8x, we get

dy/dx = e^8x (8x² - 48 + 2x)

dy/dx = 2e^8x (4x² + x - 24)

Problem 5 :

y = √x e^x

Solution :

= d/dx [√x e^x]

= d/dx [√x] e^x + √x d/dx [e^x]

= 1/2 x^1/2-1 e^x + √x e^x

dy/dx = √x e^x + e^x/2√x

Problem 6 :

y = 4e^{2x^2}

Solution :

= d/dx [4e^{2x^2}]

= 4 · d/dx [e^{2x^2}]

= 4e^{2x^2} · d/dx [2x²]

= 4e^{2x^2} · (4x)

dy/dx = 16xe^{2x^2}

Problem 7 :

y = xe^{x^2}

Solution :

= d/dx [xe^{x^2}]

u = x and v = e^{x^2}

u' = 1 and v' = e^{x^2} (2x)

= x (2xe^{x^2}) + e^{x^2} (1)

= e^{x^2} + xe^{x^2} (2x)

dy/dx = e^{x^2} + 2x²e^{x^2}

Problem 8 :

y = e^{(e^x)}

Solution :

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

= e^{(e^x)} · d/dx [e^x]

dy/dx = e^{(e^x)} · e^x

Problem 9 :

y = e^2x+1 / 2x + 7

Solution :

= d/dx [e^2x+1 / 2x + 7]

Problem 10 :

y = e^3x/x²

Solution :

= d/dx [e^3x/x²]

u = e^3x and u' = 3e^3x

v = x² and v' = 2x

= [x²(3e^3x) - e^3x (2x)] / x⁴

= e^3x (3x² - 2x) / x⁴

= e^3x (3x - 2) / x³

Problem 11 :

y = e^√x

Solution :

= d/dx [e^√x]

= e^√x · (1/2 x^1/2-1)

dy/dx = e^√x / 2√x

Problem 12 :

y = e^x + 1 / e^x - 1

Solution :