# 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

Download Practice Questions with Answers

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 = lnx^{2}

y = lnx^{2} = 2lnx

dy/dx = 2/x

3. Differentiate* y* = 2 ln (3*x*^{2} − 1)

dy/dx = 12/(3x^{2} - 1)

4. Differentiate *y* = ln(1 *− *2*x*)^{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 ****e**^{u}**:**

Examples

**1. Find the derivative of ****y**** = 10**^{3}^{x}**:**

**2. Find the derivative of ****y**** = ****e**^{x}^{2}**:**

**3. Find the derivative of ****y**** = sin(****e**^{3}^{x}**).**

**4. Find the derivative of ****y**** = ****e**^{sin }^{x}**.**

**5. Find the derivative of**

We let *u* = ln 2*x* and *v* = *e*^{2}^{x} + 2, and we'll use the derivative of a quotient formula

*u* = ln 2*x* = ln 2 + ln *x*

And for* v* = *e*^{2}^{x} + 2 we have:

Apply Quotient rule:

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

simplify...

Finally...

**6. Find the derivative of**

**Answer:**

**7. Find the derivative of**

**Answer:**

Let

then *y* = *u*^{3}.

So

and

So

## Questions

**1.**

**Answer**s:

**2.**

**Answer:**

**3.**

**Answer:**

19/27

**4. Calculate the gradient(s) of the curve at the point(s) where it crosses the given line.**

**y = 3x**^{2}** - 2x + 6, y-axis.**

**Answer:** -2

**5. The gradient of the curve y = 2x**^{2 }**+ px + q at the point (1, 3) is 9. Calculate the values of p and q.**

**Answer:** p = 5, q = -4