Gradient and basic differentiation

Learning objective:

Understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations for first and second derivatives.
Use the derivative of x^n (for any rational n), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule.
Apply differentiation to gradients, tangents and normals.

The basic concept of differentiation is denoted below:

Chain Rule: If

;

, then

Another representation is:

, so

E.g. Find

Solution 1: Let

and

Now,

and

So that,

Solution 2:

As for a brief explanation, we can just know that gradient is defined as

. From there, we can take that

. Thus, we can make equations of tangent and normal from a curve. Equation of tangent:

Equation of normal:

Another thing to add is whenever the,

, we can look for stationary points of the curve.

Find the gradient of at

A curve pass through A(4, 2) and has equation . Find the equation of the tangent to the curve at the point A.

A curve pass through the point P(5, 1) and has equation . Find the equation of normal to the curve at the point P.