3.1 Defining the Derivative

The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment $h .$
The derivative of a function $f (x)$ at a value $a$ is found using either of the definitions for the slope of the tangent line.
Velocity is the rate of change of position. As such, the velocity $v (t)$ at time $t$ is the derivative of the position $s (t)$ at time $t .$ Average velocity is given by

$v_{ave} = \frac{s (t) - s (a)}{t - a} .$

Instantaneous velocity is given by

$v (a) = s^{'} (a) = lim_{t \to a} \frac{s (t) - s (a)}{t - a} .$
We may estimate a derivative by using a table of values.

3.2 The Derivative as a Function

The derivative of a function $f (x)$ is the function whose value at $x$ is $f^{'} (x) .$
The graph of a derivative of a function $f (x)$ is related to the graph of $f (x) .$ Where $f (x)$ has a tangent line with positive slope, $f^{'} (x) > 0 .$ Where $f (x)$ has a tangent line with negative slope, $f^{'} (x) < 0 .$ Where $f (x)$ has a horizontal tangent line, $f^{'} (x) = 0 .$
If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.
Higher-order derivatives are derivatives of derivatives, from the second derivative to the $n th$ derivative.

The derivative of a constant function is zero.
The derivative of a power function is a function in which the power on $x$ becomes the coefficient of the term and the power on $x$ in the derivative decreases by 1.
The derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative.
The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.
The derivative of the difference of a function f and a function g is the same as the difference of the derivative of f and the derivative of g.
The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.
The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function.
We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative. These formulas can be used singly or in combination with each other.

Using $f (a + h) \approx f (a) + f^{'} (a) h,$ it is possible to estimate $f (a + h)$ given $f^{'} (a)$ and $f (a) .$
The rate of change of position is velocity, and the rate of change of velocity is acceleration. Speed is the absolute value, or magnitude, of velocity.
The population growth rate and the present population can be used to predict the size of a future population.
Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item.

We can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. The results are

$\frac{d}{d x} sin x = cos x \frac{d}{d x} cos x = − sin x .$
With these two formulas, we can determine the derivatives of all six basic trigonometric functions.

The chain rule allows us to differentiate compositions of two or more functions. It states that for $h (x) = f (g (x)),$

$h^{'} (x) = f^{'} (g (x)) g^{'} (x) .$

In Leibniz’s notation this rule takes the form

$\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x} .$
We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.
The chain rule combines with the power rule to form a new rule:

$If h (x) = {(g (x))}^{n}, then h^{'} (x) = n {(g (x))}^{n - 1} g^{'} (x) .$
When applied to the composition of three functions, the chain rule can be expressed as follows: If $h (x) = f (g (k (x))),$ then $h^{'} (x) = f^{'} (g (k (x)) g^{'} (k (x)) k^{'} (x) .$

The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative.
We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.

We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).
By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.

On the basis of the assumption that the exponential function $y = b^{x}, b > 0$ is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.
We can use a formula to find the derivative of $y = ln x,$ and the relationship ${log}_{b} x = \frac{ln x}{ln b}$ allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
Logarithmic differentiation allows us to differentiate functions of the form $y = g {(x)}^{f (x)}$ or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.