Worked example 1
Differentiate a polynomial
Differentiate y = 3x⁴ − 5x² + 7
- Use the power rule on each x term.
- 3x⁴ becomes 12x³
- -5x² becomes −10x
- The constant differentiates to 0.
Answer: dy/dx = 12x³ − 10x
Back to Higher Mathematics
Higher Mathematics
Differentiating powers
Differentiate polynomial terms and evaluate gradients at given x-values.
Before you start
- Use index notation confidently, including negative powers.
- Collect like terms before differentiating where useful.
- Know that constants differentiate to 0.
Explanation
Differentiation gives the gradient function. For powers of x, multiply by the power and reduce the power by 1.
At Higher, this supports tangents, normals, stationary points and optimisation.
Visual support
Method and rules
- d/dx(xⁿ) = nxⁿ⁻¹
- d/dx(axⁿ) = anxⁿ⁻¹
- d/dx(c) = 0
Worked examples
Worked example 2
Find a gradient
For y = x³ − 4x, find the gradient at x = 2.
- Differentiate: dy/dx = 3x² − 4
- Substitute x = 2
- Calculate 3(2)² − 4
So: Gradient = 8.
Worked example 3
Use a negative power
Differentiate y = 6x⁻²
- Multiply by −2.
- Reduce the power by 1.
Answer: dy/dx = −12x⁻³
Watch out
- Leaving the power unchanged.
- Forgetting that constants become 0.
- Dropping the negative sign in negative powers.
Exam reminder
Differentiation questions may ask for a gradient value, not just dy/dx. Differentiate first, then substitute the x-coordinate if a point or value is given.