Derivatives of exponential functions

About this lesson

Calculus AB (AP-aligned) › Differentiation rules

Include answers

To save a PDF, choose “Save as PDF” in the print dialog.

Derivatives of exponential functions

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curve is $y = e^x$ , the blue line its tangent at $x = a$ . Move $a$ and compare the slope of the line with the height of the point.

The curve y = eˣ and its tangent at x = a. The tangent's slope equals the height eᵃ of the point of contact.

At $a = 0$ the point is $(0, 1)$ . What is the slope of the tangent there?

1
0

Key idea

The exponent varies, so the power rule does not apply.

(e^x)' = e^x

. For any base

b > 0

:

(b^x)' = b^x \ln b

.

Worked example

$f(x) = 3e^x - 2^x$ . Find $f'(x)$ .

(3e^x)' = 3e^x

eˣ is its own derivative.

(2^x)' = 2^x \ln 2

Base 2: multiply by ln 2.

f'(x) = 3e^x - 2^x \ln 2

Combine.

Your turn, guided

Your turn: $f(x) = 5e^x + x^3$ .

f'(x) = 5e^x + 3x^2

5eˣ stays; x³ gives 3x².

Practice ★

$f(x) = 8e^x + 5x^2$ . Find $f'(x)$ .
$f(x) = 4e^x + 6x^2$ . Find $f'(x)$ .
$f(x) = 7e^x + 4x^2$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(x) = 3^x$ . Find $f'(x)$ .
$f(x) = 4^x$ . Find $f'(x)$ .

Find the slope of the tangent line to $y = 8e^x + 1$ at $x = 0$ .
Find the slope of the tangent line to $y = 2e^x + 4$ at $x = 0$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 3^x$ . Find $f'(x)$ .
$f(x) = 4^x$ . Find $f'(x)$ .
$f(x) = 2^x$ . Find $f'(x)$ .

Summary card

Exponentials

$(e^x)' = e^x$ .
$(b^x)' = b^x \ln b$ .

Watch out: $x^n$ (variable base) and $b^x$ (variable exponent) have different rules.

End of the lesson

Next lesson: Derivatives of sine and cosine Back to the chapter

Answers: Derivatives of exponential functions

A

$10 x + 8 e^{x}$
$12 x + 4 e^{x}$
$8 x + 7 e^{x}$

B

$3^{x} \ln{\left(3 \right)}$
$4^{x} \ln{\left(4 \right)}$

C

$8$
$2$

D

$3^{x} \ln{\left(3 \right)}$
$4^{x} \ln{\left(4 \right)}$
$2^{x} \ln{\left(2 \right)}$

CC BY-SA 4.0