Calculus

Find the derivative of f(x) = x³ - 4x² + 7

Step-by-step solution with explanation

Final Answer

f'(x) = 3x² - 8x

Step-by-step solution

1

Recall the Power Rule for derivatives

The Power Rule says: bring the exponent down as a multiplier, then subtract 1 from the exponent. We use this rule on each term separately.
2

Differentiate the first term x³

Bring the 3 down in front and subtract 1 from the exponent. So x³ becomes 3x².
3

Differentiate the second term -4x²

Bring the 2 down and multiply it by the coefficient -4, giving -8. Then subtract 1 from the exponent, so x² becomes x¹, which is just x.
4

Differentiate the constant term 7

Constants (plain numbers with no x) always have a derivative of 0. A flat number never changes, so its rate of change is zero.
5

Combine all terms for the final answer

Add up the derivatives of all three terms. The 0 from the constant drops out, leaving us with the final derivative.

Understanding this problem

Learning Insight

The derivative measures how fast a function is changing at any point. The Power Rule works because of how slopes of curves are built — when you zoom in on x³, the rate it grows is always tied to 3x². Constants like 7 don't contribute to change at all, so they vanish. This is why derivatives are so useful: they tell you the exact steepness of a curve at any moment.

Quick Tip

Remember 'drop it and knock it down' — DROP the exponent in front as a multiplier, then KNOCK the exponent down by 1. Do this term by term and you're done fast.

Common Mistake

A very common mistake is forgetting to multiply the dropped exponent by the existing coefficient — for example, writing -4x instead of -8x for the term -4x². Always multiply the exponent by the coefficient before reducing the power.