**Definition of derivative, relate it to the slope of the tangent of a curve, derivative of the sum, difference, product and quotient of functions**

Differentiation relates to the measurement of change. For a linear function such as y = mx + b where b is the y-intercept and m is the constant slope. Here m is defined as:

The tangent line to the graph of y = f(x) at x = c is the line through the point (c, f(c)) with slope

provided this limit exists.If the instantaneous rate of change of f(x) with respect to x exists at a point c, then it is the slope of the tangent line at that point.The derivative is the slope of the tangent line to a graph f(x) and is usually denoted f’(x).It is calculated by deducing the limit of the difference quotient as ∆x approaches 0. The process of finding the rate at which one variable changes with respect to another is called differentiation.

Notation | Meaning |

f’(x) | f prime of x |

y’ | y prime |

A noun, derivative with respect to x | |

A verb, it means to take the derivative with respect to x |

## Basic differentiation rules:

**Constant rule**

Here, c is a constant and the derivative of a constant function is zero.

Ex. f(x) = 5; f’(x)=0

**Exponent rule**

n is a real number. To find the derivative of *f *(*x*) = *x ^{n}.*Pull a copy of the exponent out in front of the term and subtract one from the exponent.

Ex. f(x) = x^{7}, f’(x) = 7x^{6}

**Constant multiple rule**

c is a constant and the derivative of a constant times a function is equal to the constant times the derivative of the function.

Ex. f(x) = 3x^{8} , f’(x) = 3(8x^{7})=24x^{7}

**Sum/difference rule**

The derivative of a sum of functions is the sum of the derivatives and the derivative of a difference of functions is the difference of the derivatives.

Ex. f(x) = 7 + x^{12 }; f’(x) = 0 +12x^{11 }=12x^{11}

**Product rule**

Ex. f(x) = (x^{3} + 2x + 5 ) (3x^{7} -8x^{2} +1)

**Quotient Rule**

Ex.

**The chain rule**

If h(x) = g(f(x)) then h’(x) = g’(f(x))∙ f’(x). Note that here h(x) is a composite function. In other words, if y = h(x) = g(u), where u = f(x), then

**The chain rule develops the general power rule.**

If h(x) = [f(x)]^{n} where n is real then,

Ex.

Ex.