BeginnerTopic 30 of 104Engineering Mathematics
Applications of derivatives
Velocity, acceleration, optimisation, related rates and linear approximation.
In depth
Derivatives give velocity and acceleration, related rates, linearisation, and curve sketching. Related rates link changing quantities through an equation, e.g. a ladder sliding or a tank draining. Linear approximation estimates small changes.
Key points
- Differentiate the relation with respect to time
- Units of derivative = output/input
- Newton's method uses derivatives
ExampleA sphere with radius growing at 2 mm/s has volume rate 4πr²·2.
Derivatives describe motion, rates and optimal designs, so they are the core tool for engineering analysis.
What is Applications of derivatives?
Velocity, acceleration, optimisation, related rates and linear approximation.
Why is it important to understand applications of derivatives?
Derivatives describe motion, rates and optimal designs, so they are the core tool for engineering analysis.
What is linear approximation?
It replaces a function near a point by its tangent line, which is accurate for small changes and makes many nonlinear problems tractable.
What do related rates problems involve?
Several quantities change in time and are linked by an equation. Differentiate that equation with respect to time to find an unknown rate.
How is Applications of derivatives used in practice?
Typical applications include finding the acceleration of a piston from its position function, estimating how fast a tank drains with a changing head and linearising a nonlinear model around an operating point.
What should I learn before Applications of derivatives?
Study "Derivatives of common functions" first (topic 29 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Applications of derivatives?
Differentiate the relation with respect to time. Units of derivative = output/input. Newton's method uses derivatives.
Can you give an example of Applications of derivatives?
A sphere with radius growing at 2 mm/s has volume rate 4πr²·2.
What comes after Applications of derivatives?
The next topic is "Maxima and minima", which builds directly on this one.
How difficult is Applications of derivatives?
It is classed as Beginner in Engineering Mathematics (topic 30 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.