Differentiation

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Description of Levels

Before beginning these exercises make sure you understand Indices really well.

Level 1 - Differentiate basic polynomials

Level 2 - Differentiate polynomials including negative and fractional indices

Level 3 - Calculations involving the gradient at the given point

Level 4 - Finding tangents and normals

Level 5 - Differentiate trigonometric functions

Level 6 - Differentiate exponential and natural logarithm functions

Level 7 - Differentiate using the chain rule

Level 8 - Differentiate using the product rule

Level 9 - Differentiate using the quotient rule

Level 10 - Interpreting derivatives and second derivatives, maxima, minima and points of inflection.

Level 11 - Differentiate simple functions parametrically

Exam Style questions are in the style of IB or A-level exam paper questions and worked solutions are available for Transum subscribers.

Integration - Exercises on indefinite and definite integration of basic algebraic and trigonometric functions.

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Example

Terminology and symbols

Please note that if $y = f(x) = x^2$ then the first differential can be shown in any of the following ways:

$$\frac{dy}{dx} = 2x$$ $$y' = 2x$$ $$f'(x) = 2x$$

Differentiating Trigonometric Functions

$$\frac{d}{dx} (\sin x) = \cos x $$ $$\frac{d}{dx} (\cos x) = -\sin x $$ $$\frac{d}{dx} (\tan x) = \frac{1}{\cos^2 x} $$

Differentiating Other Functions

$$\frac{d}{dx} (e^x) = e^x $$ $$\frac{d}{dx} ( \ln x) = \frac{1}{x} $$

In the following rules, $u$ and $v$ are functions of $x$.

The Product Rule

$$ \text{If} \quad y = uv \quad \text{then}$$ $$\frac{dy}{dx} = v\frac{du}{dx} + u\frac{dv}{dx}$$

The Quotient Rule

$$ \text{If} \quad y = \frac{u}{v} \quad \text{then}$$ $$\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$

The Chain Rule

$$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$

Parametric Equations

if $x$ and $y$ are given in terms of a third variable, the parameter, which could be $t$, then:

$$\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$$

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1. Find the second derivative of \(f\) if \(f(x) = 7x^2 - 5x - 2 \).
2. If \(y =5x^4-bx^3+4x^2-2x\) find the value of \(b\) if \( \quad \frac{d^2y}{dx^2} = 60x^2-18x+8\).
3. If \(y =6x^4-6x^3+2x^2-8x\) find the value of the second derivative, \( \frac{d^2y}{dx^2} \), when \(x=8\)
4. A function \(f\) has a derivative \(f'(x)=0 \) and a second derivative \(f''(x)=2.3 \) at the point where \(x= -1\). Which of the following features does the function have at \(x= -1\)? a. A local minimum b. A local maximum c. A point of inflection Type in the letter that represents your answer
5. The graph of \(y=x^4-3x^3+5x \) has a stationary point where \(x=1 \). By finding the first and second derivatives determine the nature of this stationary point. a. A local minimum b. A local maximum c. A point of inflection Type in the letter that represents your answer
6. Find the local minimum value of the function \(f\) if \( \quad f(x) = \frac{x^3}{3} - \frac{x^2}{2} -30x+5\)
7. If \(f(x) = e^x \) what is the value of the second derivative of \(f(x) \) when \(f(x)= 65.3\)?
8. If \(f(x) = \sin 6x \) what is the value of the second derivative of \(f(x) \) when \(f(x)= 0.29\)?
9. The function \(f\) is defined by the formula \(f(x) = 5(2x-1)^2\). By first finding the gradient function determine the value of x when the gradient is 3. Give your answer correct to three significant figures.
10. If \(y =2e^{2x}- \frac{4}{x^3}\) find the value of the second derivative, \( \frac{d^2y}{dx^2} \), when \(x=3\) Give your answer correct to three significant figures.

Differentiation

Practise the technique of differentiating polynomials and other functions with this self marking exercise.

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