Question

What does it mean to apply a derivative to a function? Also, what does it find to find the derivative with respect to a variable?

Answer 1

Think of the derivative as an operator to a function. It gives you the instantaneous slope of that function (the slope of the line tangent to the function for all x). When you take a derivative with respect of a variable all it means is that the resulting equation will give you the slope at any position for THAT variable. For instance, if you had an equation y=x^2+h^3 and solved took its derivative (assuming h is a variable), the resulting equation could only tell you the slope of the function if h is given. (dy/dx = 2x + d(h^3)/dx)

Answer 2

Derivative can also be thought as rate of change. A derivative at a certain point of a function is the rate of change of function at that point. This can also be associated with velocity. If you have the equation of movement of a body, say:

x(t) = t*0 + v0*t + (a*t^2)/2

Its derivative:

v(t) = v0 + at

Gives the instantaneous velocity at respect to time t.

Answer 3

Derivative means the "Tangent Line" and "Instantanious Rate of Change"