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9.08 Differentiation of exponential functions

Worksheet
Differentiation of exponential functions
1

Differentiate each of the following functions:

a

y = - 4 e^{x}

b

y = \dfrac{1}{2} e^{ - x }

c

y = 7 e^{ - 3 x}

d

y = 6 e^{\frac{x}{3}}

e

y = e^{ 8 x}

f

y = e^{ - 2 x } + 4

g

y = 3 e^{ 5 x} - e^{ - 4 x } + x^{2}

h

y = e^{ 3 x} + 5 x^{2} - 2

i

y = 2 e^{ 4 x} - 3 e^{ - 5 x }

j

y = e^{ 2 x} + e^{9} + e^{ - 5 x }

k

y = e^{ 5 x} + e^{ 2 x}

l

y = \dfrac{e^{x} - e^{ 3 x} + 1}{e^{x}}

m

y = \dfrac{e^{x} - e^{ - x }}{2}

n

y= e^{ 8 x} - e^{ - 3 x }

o

y = \dfrac{e^{ 9 x}}{9} + \dfrac{e^{ 8 x}}{8}

p

y = e^{x} + e^{ - x }

2

Consider the function y = e^{ a x}, where a is a constant.

a

Let u = a x. Rewrite the function y in terms of u.

b

Determine \dfrac{du}{dx}.

c

Hence determine \dfrac{dy}{dx}.

d

Consider the function f \left( x \right) = e^{ 2 x}.

i

Find f' \left( x \right).

ii

Find the exact form of f' \left( - 3 \right).

3

Consider the function f \left( x \right) = 3 - e^{ - x }.

a

Determine f \left( 0 \right).

b

Determine f' \left( 0 \right).

c

Which of the following statements is true?

A

f' \left( x \right) < 0 for x \geq 0

B

f' \left( x \right) < 0 for all real x.

C

f' \left( x \right) > 0 for all real x.

D

f \left( x \right) > 0 for all real x.

d

Determine the value of \lim_{x \to \infty} f \left( x \right).

4

Consider the function y = e^{ 2 x}.

a

Find \dfrac{d y}{d x}.

b

Find the exact value of the derivative when x = 0.

c

Find the exact value of the derivative when x = 5.

5

Consider the following functions.

i

Rewrite the function as a power of e.

ii

Find \dfrac{d y}{d x}.

a

y = \sqrt{e^{x}}

b

y = \sqrt[7]{e^{x}}

c

y = \dfrac{1}{\sqrt{e^{x}}}

Tangents and normals of exponential functions
6

Consider the function y = e^{ - x }.

a

Determine \dfrac{dy}{dx}.

b

Find the gradient of the tangent to the curve at x = - 2. Round your answer to three decimal places.

7

For each of the following functions:

i
Find the gradient of the tangent to the curve at x=1.
ii
Find the equation of the tangent to the curve at x=1.
a
y=e^{2x}
b
y=e^{-x}
c
y=5e^{x}
d
y=2xe^{x}
8

For each of the following functions:

i
Find the gradient of the normal to the curve at x=0.
ii
Find the equation of the normal to the curve at x=0.
a
y=e^{3x}
b
y=2e^{-x}
c
y=4e^{x}+x
d
y=\dfrac{x}{e^{x}}
9

Find the equation of the tangent to the curve f \left( x \right) = 2 e^{x} at the point where it crosses the y-axis. Express the equation in the form y = m x + c.

10

The tangent to the curve f \left( x \right) = 2.5 e^{x} at the point x = 7.5 is parallel to the tangent to the curve g \left( x \right) = e^{ 2.5 x} at the point where x = k. Solve for k.

11

Point R lies on the curve with equation y = e^{ 3 x + 2}.

a

If the x-coordinate of point R is - \dfrac{2}{3}, determine its y-coordinate.

b

Determine the gradient m of the tangent at the point R.

c

Find the gradient of the normal at point R.

d

Hence find the equation of the normal at the point R. Write your answer in general form.

12

Point Q \left( - 1 , e\right) lies on the curve with equation y = e^{ - x }.

a

Determine the gradient m of the tangent at the point Q.

b

Find the gradient of the normal at point Q.

c

Hence find the equation of the normal at the point Q.

d

State the coordinates of the y-intercept of the normal.

e

Determine the coordinates of the x-intercept of the normal.

f

Find the exact area in expanded form of the triangle which has vertices at these intercepts and the origin.

13

Consider the function y = 7^{x}.

a

Rewrite the function in terms of natural base e.

b

Hence determine y'. Express the derivative in terms of the base 7.

c

Hence determine the exact gradient of the tangent of the curve at x = 1.

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MA11-6

manipulates and solves expressions using the logarithmic and index laws, and uses logarithms and exponential functions to solve practical problems

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