You already find the gradient of a straight line. A curve is different: its gradient changes from point to point, and at any point it equals the gradient of the tangent drawn there. Differentiation is the process of finding the gradient function — one formula that gives the gradient of the curve at every point.
For \(y=f(x)\), the derivative is written \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=f'(x)\). Substituting the \(x\)-coordinate of a point such as \(P\) into \(f'(x)\) gives the gradient of the curve — the gradient of the tangent — at that point.
Notation
| Written as | Meaning |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=f'(x)\) | First derivative: differentiate \(y=f(x)\) with respect to \(x\). |
| \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=f''(x)\) | Second derivative: differentiate \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) with respect to \(x\) again. |
| \(\dfrac{\mathrm{d}v}{\mathrm{d}t}\) | Differentiate \(v\) with respect to \(t\) — the letters follow the variables in the question. |
| \(\dfrac{\mathrm{d}}{\mathrm{d}x}(\ \cdots\ )\) | An instruction: differentiate the expression in the brackets with respect to \(x\). |
Standard results: powers of \(x\)
| Function | Derivative | Example |
|---|---|---|
| \(y=c\), a constant | \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=0\) | \(y=8\) \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=0\) |
| \(y=kx\) | \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=k\) | \(y=\dfrac{2x}{5}=\dfrac{2}{5}x\) \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{2}{5}\) |
| \(y=x^n\), \(n\) rational — negative index | \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=nx^{n-1}\) | \(y=\dfrac{4}{x^3}=4x^{-3}\) \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=-12x^{-4}=-\dfrac{12}{x^4}\) |
| \(y=x^n\), \(n\) rational — fractional index | \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=nx^{n-1}\) | \(y=\sqrt{x}=x^{\frac{1}{2}}\) \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{1}{2}x^{-\frac{1}{2}}=\dfrac{1}{2\sqrt{x}}\) |
Constant-multiple and sum/difference rules
| Rule | In symbols | Example |
|---|---|---|
| Constant multiple | \(\dfrac{\mathrm{d}}{\mathrm{d}x}[ku]=k\dfrac{\mathrm{d}u}{\mathrm{d}x}\) | \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(4x^6\right)\) \(=4\left(6x^5\right)=24x^5\) |
| Sum / difference | \(\dfrac{\mathrm{d}}{\mathrm{d}x}[u\pm v]=\dfrac{\mathrm{d}u}{\mathrm{d}x}\pm\dfrac{\mathrm{d}v}{\mathrm{d}x}\) | \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(x^3-\dfrac{6}{x}\right)\) \(=3x^2+\dfrac{6}{x^2}\) |
Differentiating again: the second derivative
| Function | First derivative | Second derivative |
|---|---|---|
| \(y=2x^3-5x^2\) | \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=6x^2-10x\) | \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=12x-10\) |
Before differentiating, rewrite every term as a power of \(x\): \(\sqrt{x}=x^{\frac{1}{2}}\) and \(\dfrac{1}{x^n}=x^{-n}\). Differentiate term by term, then tidy negative indices back into fractions.
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