Application of Derivatives
Class 12 Mathematics · Mathematics Part I · Chapter 6
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Summary
Derivatives measure rates of change and the local behaviour of functions, and this chapter applies them widely. The derivative as a rate of change links related quantities (for example, how a sphere's volume changes with its radius). A function is increasing where \(f'(x)>0\) and decreasing where \(f'(x)<0\). The derivative also gives the slope of a tangent, so equations of tangents and normals follow at once. Maxima and minima are located using the first-derivative test and the second-derivative test: at a local maximum \(f'(x)=0\) and \(f''(x)<0\), while at a local minimum \(f'(x)=0\) and \(f''(x)>0\). These ideas drive optimisation problems — maximising area, minimising cost or material — that appear throughout applied mathematics.
Rate of change of quantitiesIncreasing and decreasing functionsTangents and normalsMaxima and minimaOptimisation problems
Key terms
Rate of change
\(\dfrac{dy}{dx}\) gives the instantaneous rate of change of \(y\) with respect to \(x\).
Increasing function
A function with \(f'(x)>0\) on an interval.
Decreasing function
A function with \(f'(x)<0\) on an interval.
Tangent slope
The slope of the tangent at a point equals \(f'(x)\) there.
Critical point
A point where \(f'(x)=0\) or \(f'(x)\) does not exist.
Second-derivative test
At a critical point, \(f''<0\) gives a maximum and \(f''>0\) a minimum.
Extra questions & answers
Try one from this chapter
A differentiable function is increasing on an interval where: