Differentiation
The derivative measures the instantaneous rate of change of a function. At a point on a curve, it gives the gradient of the tangent.
Core idea: if \(y=f(x)\), then \(f'(x)\) or \(\frac{dy}{dx}\) tells us how quickly \(y\) changes as \(x\) changes.
Essential rules
- \(\frac{d}{dx}(x^n)=nx^{n-1}\)
- \(\frac{d}{dx}(ax+b)=a\)
- \(\frac{d}{dx}(c)=0\)
Worked example
For \(f(x)=3x^4-5x^2+7\),
\[f'(x)=12x^3-10x.\]