Applications of Derivatives

Rate of Change  

Derivatives measure the instantaneous rate of change of one quantity with respect to another. In geometry, this is interpreted as the slope of the tangent to a curve at a given point.  

Equation of Tangent and Normal  

- The tangent to a curve at a point represents its slope at that point.  

- The normal is the line perpendicular to the tangent.  

- Tangents can be parallel or perpendicular to the coordinate axes, depending on the slope.  

- In parametric form, tangents and normals are expressed using the parameter defining the curve.  

Tangents from an External Point  

From a point outside a curve, tangents can be drawn by finding the points of contact on the curve.  

Lengths Related to Tangents and Normals  

At a point on a curve, we can define:  

- Length of tangent  

- Length of normal  

- Subtangent (projection of tangent on x‑axis)  

- Subnormal (projection of normal on x‑axis)  

Angle Between Curves  

The angle between two curves at their intersection is the angle between their tangents at that point. If tangents are perpendicular, the curves intersect orthogonally.  

Shortest Distance Between Curves  

For non‑intersecting curves, the shortest distance lies along their common normal.  

Errors and Approximations  

Derivatives help estimate small changes:  

- Absolute error: Actual change in a variable.  

- Relative error: Ratio of error to the actual value.  

- Percentage error: Relative error expressed as a percentage.  

Approximations use derivatives to estimate function values near a given point.  

Monotonic Functions 

- A strictly increasing function has an increasing inverse.  

- If continuous, the inverse is also continuous.  

- Composition of increasing functions remains increasing.  

- If one function increases and another decreases, their composition decreases.  

- Invertible functions must be monotonic.  

Rolle’s Theorem  

If a function is continuous on a closed interval, differentiable on the open interval, and equal at both ends, then there exists a point where its derivative is zero.  

Lagrange’s Mean Value Theorem (LMVT)  

If a function is continuous and differentiable on an interval, then there exists a point where the derivative equals the average rate of change over that interval.  

Special Points 

- Critical points: Where the derivative is zero or undefined.  

- Stationary points: Where the derivative is exactly zero.  

Every stationary point is critical, but not every critical point is stationary.  

Second Derivative and Inflexion  

- If the second derivative is positive, the curve is concave upward.  

- If negative, the curve is concave downward.  

- A point of inflexion occurs where concavity changes.  

Mensuration Applications  

Derivatives are often applied to optimise or analyse geometric quantities like surface area, volume, and curved surfaces of solids (cuboid, cube, cone, cylinder, sphere, prism, pyramid).