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Basics of differentiation for OCR A-level Maths

Basics of differentiation

This page covers the following topics:

1. Differentiating linear expressions
2. Differentiating polynomials
3. Differentiating constants
4. Differentiating fractions

The derivative of a function is the rate of change of the function with repsect to the given variable. The derivative of the linear expression x is 1. A linear expression of form ax has a derivative of a.

Differentiating linear expressions

A polynomial can be differentiated by multiplying the coefficients of each variable by their exponent and subtracting one from the exponent of each variable.

Differentiating polynomials

The Constant Rule states that the derivative of a constant function is 0.

Differentiating constants

When differentiating fractions, the given fraction should be separated if it has more than one term in its numerator. The terms of the variable in the denominator which are raised to a power must then be brought to the numerator by changing its power to its negative. If the power involves roots, it must be turned into a fractional power.

Differentiating fractions

1

Differentiate y = (xยฒ + xโต)/xโต.

y = (xยฒ + xโต)/xโต = xโปยณ + 1
dy/dx = โˆ’3xโปโด

โˆ’3xโปโด

Differentiate y = (xยฒ + xโต)/xโต.

2

Find the rate of change of the perimeter of the given shape with respect to x.

perimeter = xยฒ + xยฒ + 4 + 4 = 2xยฒ + 8
By the constant rule, dy/dx = 4x.

4x

Find the rate of change of the perimeter of the given shape with respect to x.

3

Differentiate y = 6.

By the constant rule, dy/dx = 0.

0

Differentiate y = 6.

4

Find the derivative of y = (5x + 1)/xยฒ.

y = (5x + 1)/xยฒ = 5x/xยฒ + 1/xยฒ = 5xโปยน + xโปยฒ
dy/dx = โˆ’5xโปยฒ โˆ’ 2xโปยณ

โˆ’5xโปยฒ โˆ’ 2xโปยณ

Find the derivative of y = (5x + 1)/xยฒ.

5

Find the derivative of y = 15xยฒx/(3xยณ).

y = 15xยฒx/3xยณ = 5
By the constant rule, dy/dx = 0.

0

Find the derivative of y = 15xยฒx/(3xยณ).

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