The Power Rule & Gradient Functions
One shortcut replaces every first-principles derivation: multiply by the power, drop it by one. It works for roots and reciprocals too — and differentiating twice reveals a curve's shape.
The power rule
Deriving every gradient from first principles would be exhausting. Fortunately the pattern we proved for $x^2$ and $x^3$ generalises to a single shortcut, the power rule:
In words: multiply by the power, then reduce the power by one. This works for any constant power $n$ — positive, negative, or fractional — which is what makes it so powerful. The notation $\dfrac{dy}{dx}$ (read "dee-y by dee-x") means the derivative of $y$ with respect to $x$, and is used interchangeably with $f'(x)$.
Where the rule comes from
The last lesson proved this separately for $x^2$ and $x^3$, and the pattern was hard to miss. The same argument runs for a general power $n$ — the only extra ingredient is the binomial theorem to expand $(x+h)^n$.
This is here so the rule isn't a black box — you will not be asked to reproduce it. First-principles questions only ever use small positive integer powers like $x^2$ or $x^3$, exactly as in the last lesson; for everything else you simply use the power rule. One honest caveat if you're curious: the binomial theorem expands $(x+h)^n$ for positive integer $n$, so the argument above settles that case. The rule genuinely does hold for negative and fractional powers too, but proving those needs tools you'll meet later in the course.
The trick is to rewrite every term as a power of $x$ before differentiating. A square root is a power: $\sqrt{x} = x^{1/2}$. A reciprocal is a negative power: $\dfrac{1}{x^2} = x^{-2}$. Once written this way, the power rule applies directly — for example $\dfrac{d}{dx}\!\left(x^{1/2}\right) = \tfrac12 x^{-1/2}$. Converting to index form first is the single habit that unlocks the whole topic.
Two rules for building up expressions
Together these mean you can differentiate any polynomial by handling each term separately. Two special cases fall out: the derivative of $kx$ is $k$ (a straight line has constant gradient), and the derivative of a constant is $0$ (a horizontal line has zero gradient).
The second derivative
Differentiating once gives the gradient function $f'(x)$. Differentiating again gives the second derivative, written $f''(x)$ or $\dfrac{d^2y}{dx^2}$. It measures the rate of change of the gradient — how quickly the slope itself is changing. It will be the key tool for classifying maximum and minimum points two lessons from now, and it also tells you where a curve is convex ($f''(x)>0$, curving upward) or concave ($f''(x)<0$, curving downward).
Sketching the gradient function
Because $y=f'(x)$ measures the gradient of $y=f(x)$ at every point, you can sketch one directly from a sketch of the other — without ever finding an equation. Three facts turn a picture of $f$ into a picture of $f'$:
Reading a curve left to right and tracking whether it's climbing, falling, or momentarily flat is enough to plot the sign and zeros of the gradient function — the actual steepness (how far above or below the axis $f'$ sits) is harder to pin down exactly without an equation, but the shape and sign pattern are always found this way.
Before differentiating, rewrite roots, reciprocals and fractions as powers of $x$: $\sqrt{x}\to x^{1/2}$, $\dfrac{3}{x}\to 3x^{-1}$, $\dfrac{x^2+1}{x}\to x + x^{-1}$ (split the fraction). Only once every term is $k x^n$ can you apply "multiply by the power, drop it by one" cleanly. Trying to differentiate a root or a fraction without converting is where most marks are lost.
Worked example 1
Work strictly term by term and you won't drop anything. The two easy-to-miss terms are the linear one ($7x$ differentiates to the bare constant $7$) and the constant ($-2$ differentiates to $0$, disappearing entirely). Every term of a polynomial reduces its power by exactly one.
Worked example 2
The power rule doesn't care whether $n$ is a fraction or negative: for $x^{1/2}$, one less than $\tfrac12$ is $-\tfrac12$; for $x^{-2}$, one less than $-2$ is $-3$. The arithmetic "$n-1$" runs the same regardless. Converting the final answer back into root and fraction form is good practice, but the differentiating itself happens entirely in index form.
Worked example 3
At $x=2$ the gradient is $0$ (a flat spot) and the second derivative is positive, meaning the gradient is increasing through that point — the curve is bowl-shaped, or convex. That's exactly the signature of a minimum, which the stationary-points lesson will make precise. The sign of $f''$ is the fingerprint distinguishing a valley ($f''>0$) from a peak ($f''<0$).
Worked example 4
No equation is given, and none is needed — the gradient's sign and zeros follow entirely from where $f$ is climbing, falling, or momentarily flat.
The gradient graph crosses the $x$-axis exactly at the $x$-values of $f$'s turning points — $x=-1$ and $x=2$ here — and its sign flips from $+$ to $-$ at a maximum, and from $-$ to $+$ at a minimum. This pattern holds for any curve: count the turning points of $f$, and you know exactly how many times $f'$ crosses the axis, and the sign either side of each crossing.
Two classic slips. First, forgetting to convert before differentiating: you cannot "power-rule" $\sqrt{x}$ or $\tfrac{1}{x}$ while they're still written as a root or fraction — rewrite them as $x^{1/2}$ and $x^{-1}$ first. Second, mishandling negative powers: one less than $-2$ is $-3$, not $-1$. Subtract $1$ from the power carefully, keeping track of the sign.
Check yourself
Given $y = x^4 - \dfrac{2}{x} + 5\sqrt{x}$, find $\dfrac{dy}{dx}$. Then find $\dfrac{d^2y}{dx^2}$ for $y = x^4$ only.
(Hint: rewrite $\dfrac{2}{x}=2x^{-1}$ and $5\sqrt{x}=5x^{1/2}$ before differentiating.)
Notice $-2x^{-1}$ differentiated to $+2x^{-2}$ — the two minus signs (the coefficient and the power coming down) multiplied to a plus. Sign-tracking on negative powers is exactly where care pays off.