The instantaneous rate of change of a function; the slope of its tangent line.
The derivative f'(x)=lim (f(x+h)-f(x))/h is the slope of the tangent line and the instantaneous rate of change. In physics velocity is dx/dt and acceleration d²x/dt². Differentiability requires smoothness.
If x=5t² then v=10t, so at t=2 s the speed is 20 m/s.
A derivative is a rate of change, so it gives velocity from position, current from charge and sensitivity from a calibration curve.
The instantaneous rate of change of a function; the slope of its tangent line.
A derivative is a rate of change, so it gives velocity from position, current from charge and sensitivity from a calibration curve.
The function is increasing at that point. A negative derivative means it is decreasing, and zero means it is momentarily flat.
The tangent is the best local straight-line approximation, and its slope is the limit of secant slopes as the two points merge.
Typical applications include converting position data from an encoder into speed, finding the slope of a stress-strain curve to get stiffness and computing the rate at which a tank level changes.
Study "Continuity" first (topic 26 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Notation: f', dy/dx, Df. Derivative zero at flat points. Linear approximation f(x+h)≈f(x)+f'(x)h.
If x=5t² then v=10t, so at t=2 s the speed is 20 m/s.
The next topic is "Differentiation rules", which builds directly on this one.
It is classed as Beginner in Engineering Mathematics (topic 27 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.
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