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Introduction to Derivatives

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Introduction to Derivatives - Lesson Summary

Consider f is a real valued function and a is a point in its domain of definition. The derivative of f at a is defined by(provided the limit exists):

f'(a) =  lim h → 0 f(a+h) - f(a) h

Derivative of f(x) at a is denoted by fI(a).

Find the derivative of the function, f(x) = 2x + 3 at x = 2.

f'(a) =  lim h → 0 f(a+h) - f(a) h

f'(2) =  lim h → 0 f(2+h) - f(2) h

f'(2) =  lim h → 0 f(2(2+h)+3) - f(2) h

=  lim h → 0 [4+2h+3-4-3] h

=  lim h → 0 2h h

=  lim h → 0  2

= 2

fI(2) = 2

Therefore, the derivative of the function, '2x + 3,' at x = 2, is equal to 2.

Geometrical interpretation of a derivative

Consider a function f(x). The curve here represents the function.

Let's consider a point, P, on the curve.

The coordinates of the point are 'a' and f(a).

Let Q be another point near P on the same curve as shown.

The coordinates of point Q are a+h, f(a+h).

Drop perpendiculars from points P and Q on the X- and the Y-axes.

If we consider points P and Q, the increment along the Y-axis is QR = f(a+h) - f(a).

The increment along the X-axis is RP = a + h - a = h.

According to the definition of the derivative at a point, we have the expression

f'(a) =  lim h → 0 f(a+h) - f(a) h

⇒ f'(a) =  lim Q → P QR RP

In the figure, tan QPR = QR/RP

Chord PQ tends to become a tangent at point P.

The limit is equal to the slope of this tangent.

Therefore, the derivative of the function at point P is equivalent to the tangent of the angle made by the tangent of the curve with the X-axis.

Suppose f is a real valued function. the function defined by

Wherever the limit exists, is defined to be the derivative of f at x, and is denoted by fI(x).

f'(x) =  lim h → 0 f(x+h) - f(x) h

This definition of a derivative is called the first principle of derivatives.

Ex: Find the derivative of the function, fI(x) = x2.

f'(x) =  lim h → 0 f(x+h) - f(x) h

=  lim h → 0 (x+h)2 -  (x)2 h

=  lim h → 0 x + 2xh + h2 -   x2 h

=  lim h → 0 2xh +   h2 h

=  lim h → 0 2x+h

=  lim h → 0 2x +  lim h → 0 h

fI(x) = 2x

Therefore, the derivative of the function, x2 is 2x.