§ 1.3 The Derivative

Section Outline The Derivative Differentiation Slope and the Derivative Equation of the Tangent Line to the Graph of y = f (x) at (a, f (a)) Leibniz Notation for Derivatives Calculating Derivatives Via the Difference Quotient Differentiable Limit Definition of the Derivative Limit Calculation of the Derivative

The Derivative Definition Example Derivative: The slope formula for a function y = f (x), denoted: Given the function f (x) = x3, the derivative is

Differentiation Definition Example Differentiation: The process of computing a derivative. No example will be given at this time since we do not yet know how to compute derivatives. But don’t worry, you’ll soon be able to do basic differentiation in your sleep.

Differentiation Examples These examples can be summarized by the following rule.

Differentiation Examples Find the derivative of SOLUTION This is the given equation. Rewrite the denominator as an exponent. Rewrite with a negative exponent. What we’ve done so far has been done for the sole purpose of rewriting the function in the form of f (x) = xr. Use the Power Rule where r = -1/7 and then simplify.

Differentiation Examples Find the slope of the curve y = x5 at x = -2. SOLUTION We must first find the derivative of the given function. This is the given function. Use the Power Rule. Since the derivative function yields information about the slope of the original function, we can now use to determine the slope of the original function at x = -2. Replace x with -2. Evaluate. Therefore, the slope of the original function at x = -2 is 80 (or 80/1).

Equation of the Tangent Line to the Graph of y = f (x) at the point (a, f (a))

Equation of the Tangent Line EXAMPLE Find the equation of the tangent line to the graph of f (x) = 3x at x = 4. SOLUTION We must first find the derivative of the given function. This is the given function. Differentiate. Notice that in this case the derivative function is a constant function, 3. Therefore, at x = 4, or any other value, the value of the derivative will be 3. So now we use the Equation of the Tangent Line that we just saw. This is the Equation of the Tangent Line. f (4) = 12 and Simplify.

Leibniz Notation for Derivatives Ultimately, this notation is a better and more effective notation for working with derivatives.

Calculating Derivatives Via the Difference Quotient The Difference Quotient is

Calculating Derivatives Via the Difference Quotient EXAMPLE Apply the three-step method to compute the derivative of the following function: SOLUTION STEP 1: We calculate the difference quotient and simplify as much as possible. This is the difference quotient. Evaluate f (x + h) and f (x). Simplify. Simplify. Simplify.

Calculating Derivatives Via the Difference Quotient CONTINUED Factor. Cancel and simplify. STEP 2: As h approaches zero, the expression -2x – h approaches -2x, and hence the difference quotient approaches -2x. STEP 3: Since the difference quotient approaches the derivative of f (x) = -x2 + 2 as h approaches zero, we conclude that

Differentiable Definition Example Differentiable: A function f is differentiable at x if approaches some number as h approaches zero. The function f (x) = |x| is differentiable for all values of x except x = 0 since the graph of the function has no definite slope when x = 0 (f is nondifferentiable at x = 0) but does have a definite slope (1 or -1) for every other value of x.

Limit Definition of the Derivative

Limit Calculation of the Derivative EXAMPLE Using limits, apply the three-step method to compute the derivative of the following function: SOLUTION STEP 1: This is the difference quotient. Evaluate f (x + h) and f (x). Simplify. Simplify. Simplify.

Limit Calculation of the Derivative CONTINUED Factor. Cancel and simplify. STEP 2: As h approaches 0, the expression -2x – h + 0.5 approaches -2x + 0.5. STEP 3: Since the difference quotient approaches , we conclude that