Differentiation

Basic Rules

• For a curve y = f(x), dy/dx represents the gradient function of the tangent to the curve at a point x.

• dy/dx measures the rate of change of y with respect to x.

• The derivative of f(x) = x r where r is a constant real number is given by

• f '(x) = r x r - 1

• Example

• f(x) = 3x 3 ,

• f '(x) = 9 x 2

A. Sum Rule

The derivative of f(x) = g(x) + h(x) is given by

f '(x) = g '(x) + h '(x)

B. Difference Rule

The derivative of f(x) = g(x) - h(x) is given by

f '(x) = g '(x) - h '(x)

C. Product Rule

D. Quotient Rule

E. Chain Rule

F. More complex differentiation:

Differentiation of trigo functions

Examples

1. Differentiate cos³x with respect to x.

Let y = cos³x

Let u = cos x

therefore y = u³

dy = 3u²

du

du = -sin x

dx

dy = du × dy

dx dx du

= -sin x × 3u²

= -sin x × 3cos²x

= -3cos²x sin x

Differentiation of exponential functions

Examples

1. Differentiate y = ln2x

--> 2/2x = 1/x

2. Differentiate y = lnx2

y = lnx2 = 2lnx

dy/dx = 2/x

3. Differentiate y = 2 ln (3x2 − 1)

dy/dx = 12/(3x2 - 1)

4. Differentiate y = ln(1 2x)3 = 3ln(1 - 2x)

dy/dx = -6/(1 - 2x)

If u is a function of x, we can obtain the derivative of an expression in the form eu:

Examples

1. Find the derivative of y = 103x:

2. Find the derivative of y = ex2:

3. Find the derivative of y = sin(e3x).

4. Find the derivative of y = esin x.

5. Find the derivative of

We let u = ln 2x and v = e2x + 2, and we'll use the derivative of a quotient formula

u = ln 2x = ln 2 + ln x

And for v = e2x + 2 we have:

Apply Quotient rule:

Using the derivatives we just found for u and v gives:

simplify...

Finally...

6. Find the derivative of

7. Find the derivative of

Let

then y = u3.

So

and

So

1.

2.

3.