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Chapter 15
If the function is not linear then the intuition is to compute the rate of change as the
mathematical limit value of the ratio of the differences ∆yy as ∆(xx) becomes infinitely
∆xx
small. Geometrically, this is the tangent line at (x, y = f(x)) as shown in Figure 2:
Figure 2: Rate of change for ff(xx) = xx 2 and tangential line as Δxx → 0
For instance, considering f(x) = x 2 and the derivative f'(x) = 2x in a given point, say x
= 2, we have that the derivative is positive f'(2) = 4 (see Figure 3):
Figure 3: ff(xx) = xx 2 and ff ′ (xx) = 2xx
Gradient is a generalization of the derivative for multiple variables. Note that the
derivative of a function of a single variable is a scalar-valued function, whereas
the gradient of a function of several variables is a vector-valued function. The
gradient is denoted with an upside-down delta ∇ , and called del. or nabla from the
Greek alphabet. This makes sense as delta indicates change in one variable, and the
gradient is the change in all variables.
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