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To differentiate x to the power n, multiply by n and reduce the power by 1. Free IGCSE Edexcel Maths notes with step-by-step methods and 8 FAQs.
Differentiating Integer Powers is part of Quadratic and Other Graphs in IGCSE Edexcel Higher Maths.
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IGCSE Edexcel Higher
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To differentiate x to the power n, multiply by n and reduce the power by 1. Free IGCSE Edexcel Maths notes with step-by-step methods and 8 FAQs.
Use this lesson to stay inside Quadratic and Other Graphs while connecting the detail back to the wider Topic 3: Sequences, Functions and Graphs topic.
Read the core explanations from Differentiating Integer Powers before testing yourself from memory.
This is the point where the IGCSE route becomes formally Higher-tail. Earlier rate-of-change work prepared the intuition; differentiation gives the exact gradient function.
For this first step, focus on integer powers and linear combinations. The aim is secure method and interpretation, not advanced proofs.
Worked example: Differentiating a polynomial
Differentiate y = 3x^2 + 4x - 5.
Answer: dy/dx = 6x + 4
Worked example: Finding a gradient at a point
For y = x^3, find the gradient when x = 2.
Answer: The gradient at x = 2 is 12.
Worked example: Differentiating with a Negative Coefficient
Differentiate y = 7x^3 - 4x^2 + 6x - 9 and find the gradient when x = -1.
Answer: The derivative is 21x^2 - 8x + 6, and the gradient at x = -1 is 35.
Differentiate first, then substitute. If you substitute into the original function before differentiating, you only get a y-value, not a gradient.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
The derivative is the result of differentiation: a new function that gives the gradient of the original function at each point. Substituting an x-value into the derivative gives the gradient there, not the y-value.
Differentiation is finding a new function that gives the gradient of the original function. The result is called the derivative, and for integer powers it uses the power rule.
To differentiate x to the power n, multiply by n and reduce the power by 1. Differentiate each term separately, and a constant disappears because its derivative is zero.
Differentiate each term: d/dx(3x²) = 6x, d/dx(4x) = 4 and the constant -5 differentiates to 0. So if y = 3x² + 4x - 5, then dy/dx = 6x + 4.
Differentiating gives dy/dx = 21x² - 8x + 6. Substituting x = -1 gives 21(-1)² - 8(-1) + 6 = 21 + 8 + 6 = 35, so the gradient at x = -1 is 35.
The derivative of ax is a. For example, d/dx(4x) = 4, because the power of x drops from 1 to 0.
The gradient of y = x³ when x = 2 is 12. Differentiating gives dy/dx = 3x², and substituting x = 2 gives 3(2²) = 12, the gradient of the tangent there.
To find a gradient, substitute into the derivative, not the original curve. Substituting into the original function only gives a y-value, so the method is to differentiate first, then substitute.