Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey
ISBN #9781292108971
11th Edition
3,612 Questions
Homework Questions
This section introduces a variety of differentiation techniques designed to simplify the process of calculating derivatives. Key rules including the constant, power, constant multiple, sum/difference, and product rules are explained and applied to both theoretical examples and real-world problems. Understanding and applying these rules not only streamline the derivation process but also give insights into economic and scientific applications where the rate of change is crucial.
1
Explain the concept of the derivative and its importance in measuring rates of change.
2
Demonstrate how to apply differentiation rules including the constant, power, constant multiple, and sum/difference rules.
3
Apply the product rule to differentiate functions that are products of two or more functions.
4
Solve real-world problems involving marginal analysis in business and economics using derivative techniques.
CONCEPT
DEFINITION
Derivative
A measure of the rate of change or the slope of the tangent line to a function at a given point. It can be represented as f'(x) or dy/dx.
Constant Rule
The derivative of a constant function is 0 because constant functions do not change.
Power Rule
For any function f(x) = x^n (n any real number), the derivative is f'(x) = n*x^(n-1).
Constant Multiple Rule
The derivative of a constant times a function is equal to the constant multiplied by the derivative of the function; that is, d/dx [k*g(x)] = k*g'(x).
Sum or Difference Rule
The derivative of a sum or difference of functions is the sum or difference of their derivatives: d/dx[u(x) ± v(x)] = u'(x) ± v'(x).
Product Rule
For two functions u(x) and v(x) that are differentiable, the derivative of their product is given by u(x)*v'(x) + v(x)*u'(x).
Marginal Analysis
An application of derivatives in economics to approximate the rate of change in cost, revenue, or profit with respect to production levels.
Find the derivative of each function defined as follows. $$y=x^{3}-11 x^{2}+7 x+9$$
Find the derivative of each function defined as follows. $$y=8 x^{3}-5 x^{2}-\frac{x}{12}$$
Find the derivative of each function defined as follows. $$y=x^{3}-\frac{x^{2}}{16}+4 x+9$$
Find the derivative of each function defined as follows. $$y=5 x^{4}+9 x^{3}+12 x^{2}-7 x$$
Find the derivative of each function defined as follows. $$f(x)=6 x^{3.5}-10 x^{0.5}$$
QUESTION
Find the derivative of y = x^3 using the power rule.
STEP-BY-STEP ANSWER:
Step 1: Identify the function and the exponent. Here, y = x^3 and n = 3. Step 2: Apply the power rule: the derivative is n*x^(n-1). Step 3: Multiply 3 by x^(3-1), which gives 3x^2. Final Answer: The derivative is y' = 3x^2.
Power Rule Example
Find the derivative of y = 8x^4.
Step 1: Identify the constant (8) and the function part (x^4). Step 2: Apply the constant multiple rule: The derivative is 8 times the derivative of x^4. Step 3: Use the power rule for x^4, which is 4x^(4-1) = 4x^3. Step 4: Multiply 8 by 4x^3 to get 32x^3. Final Answer: The derivative is y' = 32x^3.
Constant Multiple Rule Example
If u(x) = 2x + 3 and v(x) = 3x^2, find the derivative of the product f(x) = u(x)*v(x).
Step 1: Compute the derivatives: u'(x) = 2 and v'(x) = 6x. Step 2: Apply the product rule: f'(x) = u(x) * v'(x) + v(x) * u'(x). Step 3: Substitute the corresponding functions: f'(x) = (2x + 3)*(6x) + (3x^2)*(2). Step 4: Multiply out the terms: (2x + 3)*6x = 12x^2 + 18x and 3x^2*2 = 6x^2. Step 5: Sum the results: 12x^2 + 18x + 6x^2 = 18x^2 + 18x. Final Answer: The derivative is f'(x) = 18x^2 + 18x.
Product Rule Example