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The limit of a function means finding the value of a curve at a particular point. But what is that point? A value to many decimal places can be very long, to define a value that is very precise. The precise point can be infinitely small. We can only ask...
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If there is a relationship between two or more variables (like, area and radius of a circle (A = πr2 ), or pressure, volume and temperature of a gas), then there will also be a relationship between how these variables change. You may need to find how fast one variable...
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Sometimes a small change in one variable can render a big change in a larger value. For example, a small increase (or error) in the radius of a sphere means a lot more volume is added! If you estimate the small error in one variable, you can calculate the significant...
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Implicit differentiation enables you to find the derivative of y with respect to x without having to solve the original equation for y. If we are able to write an equation relating \(x\) and \(y\) explicitly, that is in the form \(y=f(x)\), then we can find the derivative function \(y=f'\left(x\right)\)...
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Partial derivatives reveal how a function with many variables changes when you adjust just one of the variables in the input. Let us suppose that we have the equation for a paraboloid with an elliptical cross-section such as \(z=x^{2}+4y^{2}.\) In this case we have a function of two independent variables,...
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Higher derivatives are used in many mathematical, scientific and engineering subjects. Consider a function \(y=f(x)=4x^{3}-6x^{2}+7\). If we differentiate this function we obtain \(f^{\prime}\left(x\right)=12x^{2}-12x\). This is the first derivative of the function \(y=f\left(x\right)\). It is possible to find second, third and subsequent derivatives by continuing to differentiate and these are called...
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Sometimes it is easier to differentiate the logarithm of a function than the original function. This is called logarithmic differentiation and this module provides an overview of the method and provides some examples. Suppose you have to differentiate \[\begin{align*} y & =\frac{x^{2}-1}{x^{8}\sqrt{x^{4}+1}}. \end{align*}\] At first sight, you need to use...
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A tangent is a line that touches a curve at only one point. Where that point sits along the function curve, determines the slope (i.e. the gradient) of the tangent to that point. A derivative of a function gives you the gradient of a tangent at a certain point on...
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Learn how to take a derivative of a function using first principles. Using this method is the best way to understand the concepts around differentiation. Start here to really appreciate what you are doing when you differentiate, before you start differentiating using other methods in later modules. Definition The derivative...
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Learn about the rules for differentiation and the different notations that are used. This section includes algebraic, exponential, logarithmic and trigonometric examples. It is not always convenient to use differentiation from first principles to find a derivative function. The “rules” shown below have been established from first principles and can...
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How do you differentiate a larger function that has components that are smaller functions? This module will show you how to package these functions and work them out separately, before plugging them into the Chain Rule formula. The derivatives of functions such as \(y=\sin\left(x^{3}\right)\)and \(f(x)=\left(x^{2}-1\right)^{4}\) can be found using the...
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What is the product rule? It is useful when you want to differentiate a function that comprises one function multiplied by another function. The derivatives of functions such as \(y=f(x)=2x\sin\left(x\right)\) and \(y=f(x)=xe^{x}\) can be found using the product rule. Definition If \[\begin{align*} y & =f\left(x\right)\\ & =u\left(x\right)\cdot v\left(x\right) \end{align*}\] then...
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What is the quotient rule? The quotient rule is like the product rule but this time it is for one function that is divided by another (rather than multiplied). Review this section to learn how to differentiate using the quotient rule. Functions such as \(y=f(x)=\frac{1}{x^{2}+x}\), \(y=f(x)=\frac{\sin x}{x}\) and \(y=f(x)=\frac{x^{2}+1}{x+1}\) may...
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How do you find the maximum (highest) or minimum (lowest) value of a curve? The maximum or minimum values of a function occur where the derivative is zero. That is where the graph of the function has a horizontal tangent. If you go looking for the horizontal tangents (i.e. where...
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If you find some key points of a function such as: maxima, minima, or turning points; x and y axis intercepts; and regions where the gradient is positive or negative, you can put together a sketch of a curve. Read this section to find examples of this being done. To...
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The value of a function changes when one of the variables (x, y, a, or b etc) changes. It may change like the variable, (both doubling) or it may change more, or less. Differentiation means finding the derivative of an expression. This means you are finding a rate of change...
