# Differentiation

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

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This is level 10: interpreting derivatives and second derivatives, maxima, minima and points of inflection. You can earn a trophy if you get at least 7 questions correct and you do this activity online.

You may use a scientific calculator but not a GDC for this exercise

 1. Find the second derivative of $$f$$ if $$f(x) = 6x^2 - 2x - 7$$. 2. If $$y =4x^4-bx^3+2x^2-5x$$find the value of $$b$$ if $$\quad \frac{d^2y}{dx^2} = 48x^2-18x+4$$. 3. If $$y =6x^4-5x^3+5x^2-7x$$find the value of the second derivative, $$\frac{d^2y}{dx^2}$$, when $$x=9$$ 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 minimumb. A local maximumc. A point of inflectionType 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 minimumb. A local maximumc. A point of inflectionType 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 $$f(x)= 49.6$$? 8. If $$f(x) = \sin 6x$$ what is the value of the second derivative of $$f(x)$$ when $$f(x)= 0.36$$? 9. The function $$f$$ is defined by the formula $$f(x) = 3(4x-1)^2$$. By first finding the gradient function determine the value of x when the gradient is 9.Give your answer correct to three significant figures. 10. If $$y =4e^{3x}- \frac{3}{x^3}$$find the value of the second derivative, $$\frac{d^2y}{dx^2}$$, when $$x=3$$Give your answer correct to three significant figures.
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This is Differentiation level 10. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Level 7 Level 8 Level 9 Level 11

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## Description of Levels

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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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## 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:

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