Differentiate f(x) = x cubed plus 4x squared minus 7x plus 2

x cubed gives 3x squared. 4x squared gives 8x. minus 7x gives minus 7. The constant 2 gives zero. 3x squared plus 8x minus 7

Differentiate f(x) = 5x to the fourth power

The coefficient 5 stays and multiplies. Bring the exponent 4 down: 4 times 5 is 20. Reduce the exponent by one. 20x cubed

Find the slope of the tangent to f(x) = x squared minus 6x at x = 4

Differentiate: 2x minus 6. The slope at a point is the derivative there. Substitute 4: 8 minus 6. The slope is 2

Introduction to Derivatives · Grade 12

How do you differentiate a polynomial?

Use the power rule on each term: bring the exponent down in front as a multiplier and reduce the exponent by one. A coefficient comes along unchanged, a sum is differentiated term by term, and a constant term differentiates to zero. The result is a polynomial one degree lower.

Learning objectives

Differentiate a power using the power rule
Differentiate a sum, a difference, and a function multiplied by a constant
Differentiate a polynomial function and evaluate its derivative at a given point
Differentiate a function containing a root or a negative power by first writing it as a power

The power rule

For a term that is x raised to a power, the derivative is that power written in front as a multiplier, with the power itself reduced by one. x cubed differentiates to three x squared.

The rule is not an arbitrary recipe — it is what the limit from the previous lesson produces for every whole power, and it is worth deriving once for x squared to see that.

It works for negative and fractional powers too, which is what makes it enough for roots and reciprocals later, once those are rewritten as powers.

Sums, differences and coefficients

A sum differentiates term by term, and so does a difference. There is no interaction between terms, so a long polynomial is just several short problems.

A constant multiplier stays put and multiplies the result. A constant term on its own differentiates to zero, because a constant does not change and a rate of change of something that does not change is zero.

Why you check the degree

Each differentiation lowers the degree by exactly one. A cubic differentiates to a quadratic, a quadratic to a linear function.

So a quick look at the degree catches a whole class of errors before any substitution. If a cubic has differentiated to something still cubic, a term was left alone.

Worked examples

Differentiate f(x) = x cubed plus 4x squared minus 7x plus 2
1. x cubed gives 3x squared
2. 4x squared gives 8x
3. minus 7x gives minus 7
4. The constant 2 gives zero
Answer: 3x squared plus 8x minus 7
Differentiate f(x) = 5x to the fourth power
1. The coefficient 5 stays and multiplies
2. Bring the exponent 4 down: 4 times 5 is 20
3. Reduce the exponent by one
Answer: 20x cubed
Find the slope of the tangent to f(x) = x squared minus 6x at x = 4
1. Differentiate: 2x minus 6
2. The slope at a point is the derivative there
3. Substitute 4: 8 minus 6
Answer: The slope is 2

Common mistakes

Leaving the constant term in place: A constant does not change, so its rate of change is zero. Carrying it through is the most common slip in this topic, and checking the degree usually reveals it.
Forgetting to bring the exponent down: The power rule does two things: it multiplies by the old exponent and it lowers it. Doing only the second gives x squared from x cubed instead of three x squared.
Differentiating a product term by term: Sums differentiate term by term; products do not. A product needs the product rule, which is the next lesson, and treating it as a sum gives an answer that is simply not the derivative.

What to remember

Bring the power down, reduce it by one.
Coefficients stay; constants go to zero.
Sums differentiate term by term.
The degree must drop by exactly one.