Complete the following table of values to determine the value of \lim_{h \to 0}\left(\dfrac{e^{h} - 1}{h}\right). Write all values to four decimal places.

h	-0.1	-0.01	-0.001	0.001	0.01	0.1
\dfrac{e^h-1}{h}

What is the limiting value of \dfrac{e^{h} - 1}{h} as h approaches 0?

Hence, state f' \left( x \right).

Approximate \dfrac{a^{h} - 1}{h} for different values of a by filling in the gaps in the table below. Round your answers to six decimal places where necessary.

h	a=2	a=e	a=5
1	1	1.718\,282
0.1
0.01	0.695\,555	1.005\,017	1.622\,459
0.001	0.693\,387	1.000\,500	1.610\,734
0.0001	0.693\,171	1.000\,050	1.609\,567
0.000\,01

State the values of a for which: \lim_{h \to 0}\left(\dfrac{a^{h} - 1}{h}\right) = 1

The value of e^{x}, where e is the natural base, can be given by the expression below:

e^{x} = 1 + \dfrac{x}{1!} + \dfrac{x^{2}}{2!} + \dfrac{x^{3}}{3!} + \dfrac{x^{4}}{4!} + \dfrac{x^{5}}{5!} + \text{. . .}

Find the derivative of the expression above to show \dfrac{d}{dx} \left(e^{x}\right)=e^x.

Differentiate each of the following functions:

y = - 4 e^{x}

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

y = 7 e^{ - 3 x}

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

y = e^{ 8 x}

y = e^{ - 2 x } + 4

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

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

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

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

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

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

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

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

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

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

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

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

Determine \dfrac{du}{dx}.

Hence determine \dfrac{dy}{dx}.

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

Find f' \left( x \right).

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

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

If f \left( 4 \right) = 54.59815, state f' \left( 4 \right) correct to five decimal places.

If f \left( - 5 \right) = 0.00674, state f' \left( - 5 \right) correct to five decimal places.

Consider the function y = e^{ 3 x}.

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

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

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

Consider the following functions.

Rewrite the function as a power of e.

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

y = \sqrt{e^{x}}

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

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

Differentiate each of the following functions:

y = e^{ \frac{1}{6} x^{6}}

y = 3 e^{ 3 x^{2}}

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

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

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

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

y = 4 e^{\sqrt{x}}

y = \left(e^{x} - 4\right)^{4}

y = \left( 2 x - e^{ 3 x}\right)^{7}

y = \left(e^{ 2 x} + 5 x\right)^{\frac{3}{4}}

Differentiate the following functions:

y = x^{5} e^{ 4 x}

y = x^{5} e^{ - 6 x }

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

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

y = \dfrac{e^{ 3 x}}{5 x}

y = e^{x} \left(x^{4} + 1\right)

y = e^{ 2 x} \left(x^{3} + 4 x + 5\right)

y = \left(x^{2} + 8 x + 2\right) e^{ - x }

y = \sqrt{x} e^{ - 3 x }

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

y = \dfrac{e^{ 6 x}}{1 + e^{x}}

y = \dfrac{e^{x}}{e^{ 6 x} + 7}

y = \dfrac{e^{x} - 7}{e^{x} + 7}

Gradients, tangents and normals

For the function y = e^{x} at the value of x given below:

State the exact gradient of the curve.

State the gradient of the curve to two decimal places.

x=1

x=7

x=-1

x=\dfrac{1}{2}

Find the gradient of the tangent to the curve y = 7 e^{x} at the point where x = 1.3. Round your answer to two decimal places.

Find the gradient of the tangent to the curve y = - e^{x} at the point \left(1, - e \right).

State the value of x where the gradient of the tangent to the curve y = e^{x} is \dfrac{1}{e^{3}}.

Consider the function y = e^{x}-1.

Point P lies on the curve and its x-coordinate is 4, state the y-coordinate of P.

What is the gradient of the tangent at point P?

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

Determine \dfrac{dy}{dx}.

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

Consider the function f \left( x \right) = 4 x e^{ 4 x}.

Determine f' \left( x \right).

Determine the gradient of f \left( x \right) at the point x = - 1.5. Leave your answer in exact form.

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

Consider the curve with equation y = e^{x}.

Determine the gradient of the tangent to the curve at the point Q \left( - 1 , \dfrac{1}{e}\right).

Hence, find the equation of the tangent to the curve at point Q.

Does this tangent line pass through the point R \left( - 2 , 0\right)?

Consider the function y = e^{x}.

Find the gradient of the tangent at the point indicated.

Hence, find the angle of inclination of the tangent at this point. Give your answer in degrees rounded to two decimal places where necessary.

At the point where x = 0.

At the point where x = 5.

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

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

Determine the gradient of the tangent at the point R.

Find the gradient of the normal at point R.

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

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

Determine f \left( 0 \right).

Determine f' \left( 0 \right).

Is f'(x) positive or negative for all real x?

Find \lim_{x \to \infty} f \left( x \right).

For each of the following functions:

Find the gradient of the tangent to the curve at x=1.

Find the equation of the tangent to the curve at x=1.

y=e^{2x}

y=e^{-x}

y=5e^{x}

y=2xe^{x}

For each of the following functions:

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

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

y=e^{3x}

y=2e^{-x}

y=4e^{x}+x

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

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.

Find the equation of the normal to the curve f \left( x \right) = 4- e^{\frac{x}{2}} at the point where it crosses the y-axis.

Consider the curve f \left( x \right) = e^{x} + e x. Show that the tangent to the curve at the point \left(1, 2 e\right) passes through the origin.

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.

Find the value of k.

Consider the functions g \left( x \right) = e^{ 7 x} and f \left( x \right) = e^{ - 7 x }.

Given:

y_1 is the tangent line to g \left( x \right) at x = 0.
y_2 is the tangent line to f \left( x \right) at x = 0.

Determine the acute angle between y_1 and y_2 to one decimal place.

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

Find the equation of the normal at the point Q.

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

Outcomes

3.2.1.1

estimate the limit of 𝑎^(ℎ−1)/ℎ as ℎ→0 using technology, for various values of 𝑎>0

3.2.1.2

recognise that 𝑒 is the unique number 𝑎 for which the limit (in 3.2.1.1) is 1

3.2.1.4

establish and use the formula 𝑑/𝑑x (𝑒^𝑥)=𝑒^𝑥 and 𝑑/𝑑x 𝑒^[𝑓(𝑥)] = 𝑓′(𝑥)𝑒^𝑓(𝑥)

3.2.4.1

select and apply the product rule, quotient rule and chain rule to differentiate functions; express derivatives in simplest and factorised form

2.05 Differentiation and exponential functions

Outcomes

3.2.1.1

3.2.1.2

3.2.1.4

3.2.4.1

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