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- Maths and statistics
- Differentiation
You can sketch an accurate graph of a function if you know some of its key characteristics. Use this resource to learn how to bring together everything to sketch a curve. To sketch a curve, it is helpful to know the key points of the function, like: maxima, minima, or...
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Differentiation is a fundamental concept in calculus that deals with finding the rate at which things change. It is used to calculate slopes of curves, optimise functions in economics, and model motion in physics. Differentiation helps us understand how quantities vary, from tracking speed to finding maximum profit. Use this...
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- Maths and statistics
- Differentiation
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. Derivative of a function The derivative of a function \(f(x)\) is denoted by \(f'(x)\). It is defined as: \[f'(x)=\lim_{h\rightarrow0}\left(\frac{f(x+h)-f(x)}{h}\right)\quad h\neq0\] Using this definition is called...
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- Maths and statistics
- Differentiation
Gradients and tangents help us understand how functions change. We use derivatives to find these. Learn how these concepts work together to analyse and interpret the behaviour of functions. Tangent at a point along a curve A tangent is a line that touches a curve at only one point. Where...
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- Maths and statistics
- Differentiation
Higher order derivatives reveal even deeper insights into the behaviour of functions, such as jerk in motion or the curvature of graphs. These concepts are important in fields like physics, engineering and economics, where understanding complex dynamics is crucial. Higher order derivatives are derivatives of derivatives, providing insights into how...
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- Maths and statistics
- Differentiation
Implicit differentiation enables you to find the derivative of y with respect to x without having to solve the original equation for y. Implicit functions Sometimes, equations involve variables that are intertwined and not easily separated. For example, \(y^{5}+3xy+x^{2}-5=0\) or \(y=\sin(xy))\). In such expressions, \(y\) is said to be an...
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- Maths and statistics
- Differentiation
Limits help us explore how functions behave as they approach specific points or infinity, providing the foundation for concepts like continuity and derivatives. You need this skill in many real-world applications, from engineering to physics, where predicting behaviour is crucial. Use this resource to learn how to calculate and interpret...
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- Maths and statistics
- Differentiation
We can use derivatives to estimate how a function behaves around a certain point. With linear approximation, we can analyse these small variations and predict values. These methods are useful in fields like physics and engineering, where precise measurements and predictions are crucial, but often challenging to obtain directly. Sometimes...
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- Maths and statistics
- Differentiation
Logarithmic differentiation makes it easier to differentiate complex functions. This method is particularly useful for functions with variable exponents or intricate products. You might encounter logarithmic differentiation when calculating compound interest, analysing population growth models or optimising processes in engineering and physics. Sometimes it is easier to differentiate the logarithm...
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- Maths and statistics
- Differentiation
Taking the derivative of the function lets us find critical points where the slope is \(0\) or undefined. These points tell us where the function might have a maximum or a minimum value. This approach is useful in many real-world situations, like optimising resources or finding the best design for...
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- Maths and statistics
- Differentiation
Partial derivatives help us analyse the influence of each individual variable on a system. This is crucial for understanding surfaces and optimisation problems in fields like engineering, physics and economics. Use this resource to learn how to find partial derivatives. Partial derivatives reveal how a function with many variables changes...
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- Maths and statistics
- Differentiation
If there is a relationship between two or more variables, then there will also be a relationship between how these variables change. You may need to find how fast one variable changes in relation to another variable that is changing. This is called the rate of change. We often study...
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- Maths and statistics
- Differentiation
Operational rules for differentiation allow us to handle complex expressions by breaking them down into simpler parts. They are used in real-world applications, such as calculating the speed of a moving car or determining the growth rate of a population. Understanding these rules makes it easier to calculate derivatives in...
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- Maths and statistics
- Differentiation
The chain rule helps us differentiate larger functions composed of smaller functions. Use this resource to learn how to apply the chain rule. Video tutorial – using the chain rule Watch this video to learn how to use the chain rule to find the derivative of a function. The chain...
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- Maths and statistics
- Differentiation
The product rule helps us differentiate functions that are one function multiplied by another. Use this resource to learn how to apply the product rule. Video tutorial – using the product rule Watch this video to learn how to use the product rule to find the derivative of a function....
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- Maths and statistics
- Differentiation
The quotient rule helps us differentiate functions that are one function divided by another. Use this resource to learn how to apply the quotient rule. Video tutorial – using the quotient rule Watch this video to learn how to use the quotient rule to find the derivative of a function....
