Topics
6. Differential Calculus
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AustraliaNSW
Year 12

6.01 Differentiating exponential functions

Lesson
Worksheet
Practice
Worksheet
Differentiate exponential functions with base e
1

Differentiate the following:

a
y = e^{x^{2} + 5}
b
y = e^{x^{2} + 3 x + 5}
c
y = e^{ 4 x^{2} - 3 x}
d
y = 3 e^{ 5 x} - e^{ - 4 x } + x^{2}
e
y = e^{ 3 x} + 5 x^{2} - 2
f
y = e^{ \frac{1}{6} x^{6}}
g
y = 3 e^{ 3 x^{2}}
h
y = 3 e^{x^{2} - 3 x + 4}
i
y = 4 e^{\sqrt{x}}
2

Differentiate the following, expressing your answer in factorised form where possible:

a
y = \dfrac{x}{e^{x}}
b
y = \dfrac{e^{ 3 x}}{5 x}
c
y = \sqrt{x} e^{ - 3 x }
d
y = \dfrac{e^{x} - e^{ 3 x} + 1}{e^{x}}
e
y = \dfrac{e^{ 6 x}}{1 + e^{x}}
f
y = \dfrac{e^{x}}{e^{ 6 x} + 7}
g
y = \dfrac{e^{x} - 7}{e^{x} + 7}
h
y = \dfrac{e^{ 3 x} + 5}{e^{ 3 x} - 5}
i
y = e^{ 3 x - 4} \left(x + 2\right)^{2}
j
y = e^{ - 4 x } \sqrt{x + 2}
k
y = \dfrac{e^{ 2 x}}{x}
l
y = x^{5} e^{ 4 x}
m
y = x^{5} e^{ - 6 x }
n
y = e^{x} \left(x^{4} + 1\right)
o
y = e^{ 2 x} \left(x^{3} + 4 x + 5\right)
p
y = \left(x^{2} + 8 x + 2\right) e^{ - x }
q
y = x^{3} + x^{2} e^{ 3 x}
3

Differentiate the following:

a
y = \left(e^{x} - 4\right)^{4}
b
y = \left( 2 x - e^{ 3 x}\right)^{7}
c
y = \left(e^{ 2 x} + 5 x\right)^{\frac{3}{4}}
d
f(x)=\left(e^x +4\right)^5
e
f(x)=\left(e^{2x} + 3x \right)^6
f
f(x)=\left(e^{4x} + x^2\right)^4
g
f(x)=\left(5e^{6x} -9 \right)^7
h
f(x)=\left(16 - e^{2x}\right)^3
i
f(x)=\left(5x+e^{6x} \right)^7
j
f(x)=\left(e^{x^2} - 8 \right)^4
k
f(x)=\dfrac{1}{\left(e^{x} + 1 \right)^6}
l
f(x)=\dfrac{e^{2x}}{\left(e^{x} + 5 \right)^4}
4

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

a

Determine f' \left( x \right).

b

Evaluate f' \left( 0 \right).

5

Consider the function f \left( x \right) = e^{ - 3 x } \left(x^{5} + 6 x^{2} + 4\right). Evaluate f' \left( 1 \right).

Differentiate exponential functions with other bases
6

Consider the function y = 7^{x}.

a

Using the fact that e^{\ln a} = a, rewrite the function in terms of natural base e.

b

Determine y', in terms of the base 7. You may use the substitution u = \left(\ln 7\right) x.

c

Hence, determine the exact gradient at x = 1.

7

Differentiate the following functions:

a
f \left(x\right) = 5^{x}
b
f \left(x\right) = 2 \left(7^{x}\right)
c
f \left(x\right) = - 4^{x} + 3
d
f \left(x\right) = \dfrac{x^{3} - 1}{2^{x}}
e
f \left(x\right) = x^{2} \left(5^{x}\right)
f
f \left(x\right) = 2^{ - t }
g
P \left( t \right) = 2^{ 3 t}
h
f \left(x\right) = 5^{2 - 4 x} + 3
Applications
8

Consider the function f(x) = e^{2x} + 6x-8.

a

Find the gradient of the tangent to the curve at x = 0.

b

Find the equation of the tangent to the curve at x = 0.

9

Consider the function f(x) = \left(e^{2x} - 2\right)^3.

a

Find the gradient of the normal to the curve at x = 0.

b

Find the equation of the normal to the curve at x = 0.

10

Find the equation of the tangent to the curve y=2xe^{3x}+x^2 - 8 at the point (0,-8) .

11

Find the equation of the normal to the curve y=\dfrac{6}{e^{6x}-4} at the point (0,-2) .

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Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-6

applies appropriate differentiation methods to solve problems

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