AE 13: Derivation
In this exercise, we will:
Practice Basic Differentiation:
- Apply the power rule to find the derivatives of simple polynomial functions.
Apply the Chain Rule and Product Rule:
- Use these rules to differentiate more complex functions involving compositions and products.
Solve Advanced Differentiation Problems:
- Tackle derivatives of functions with nested compositions and multiple variables, reinforcing the use of the chain rule and product rule.
Common functions + their Derivatives
- Power Function:
- Function:
\[ f(x)=x^n \]
- Derivative:
\[ f'(x) = \frac{d}{dx} (x^n) = nx^{n-1} \]
- Exponential Function:
- Function:
\[ f(x)=e^x \]
- Derivative:
\[ f'(x) = \frac{d}{dx} (e^x) = e^x \]
- Function:
\[ f(x)=a^x \]
- Derivative:
\[ f'(x) = \frac{d}{dx} (a^x) = a^x \ln(a) \]
- Natural Logarithm
- Function:
\[ f(x) = ln(x) \]
- Derivative:
\[ f'(x) = \frac{d}{dx} (\ln(x)) = \frac{1}{x} \]
- Trigonometric Functions
- Function:
\[ f(x)=\sin(x) \]
- Derivative:
\[ f'(x) = \frac{d}{dx} (\sin(x)) = \cos(x) \]
- Function:
\[ f(x)=\cos(x) \]
- Derivative:
\[ f'(x) = \frac{d}{dx} (\cos(x)) = -\sin(x) \]
- Function:
\[ f(x)=\tan(x) \]
- Derivative:
\[ f'(x) = \frac{d}{dx} (\tan(x)) = \sec^2(x) \]
- Hyperbolic Functions
- Function:
\[ f(x)=\sinh(x) \]
- Derivative:
\[ f'(x) = \frac{d}{dx} (\sinh(x)) = \cosh(x) \]
- Function:
\[ f(x)=\cosh(x) \]
- Derivative:
\[ f'(x) = \frac{d}{dx} (\cosh(x)) = \sinh(x) \]
Derivatives
For each problem, find the derivative of the given function. Show all steps clearly.
Exercise 1:
Function:
\[ f(x) = 5x^3 \]
Solution:
The power rule states that \(\frac{d}{dx}(x^n)=nx^{n-1}\)
Applying the power rule:
add response here.
Exercise 2
Function:
\[ g(x) = \sqrt{x} \]
Solution:
Rewrite the function with a fractional exponent: \(\sqrt{x}=x^{1/2}\).
Apply the power rule:
add response here.
Exercise 3
Function:
\[ h(x)=\ln(x) \]
Solution:
- The derivative of the natural logarithm function is ___
add response here.
Intermediate Derivatives Using Chain Rule and Product Rule
Exercise 4
Function:
\[ f(x)=(2x^3+3x)^4 \]
Solution
Identify the outer function and inner function:
Outer function: add response here.
Inner function: add response here.
Apply the chain rule:
add response here.
- Differentiate the inner function:
add response here.
- Combine the results:
add response here.
Exercise 5
Function:
\[ \frac{d}{dx}(x^2 e^x) \]
Solution:
Identify the product of two functions: ___ and ___
Apply the product rule: \((u \cdot v)^{'}=u^{'} \cdot v + u \cdot v^{'}\)
Differentiate each function:
add response here.
- Combine the results:
add response here.
Exercise 6
Function:
\[ h(x)=\sin (x^2) \]
Solution:
Identify the outer function and inner function:
Outer function: add response here.
Inner function: add response here.
Apply the chain rule: add response here.
Differentiate the inner function:
add response here.
- Combine the results:
add response here.
Advanced Derivatives
Exercise 7
Function:
\[ f(x)=(\ln (x) \cdot e^{2x})^3 \]
Solution
Identify the outer function and inner function
Outer function: add response here.
Inner function: add response here.
Apply the chain rule: add response here.
Use the product rule to differentiate the inner function:
add response here.
- Differentiate each function:
add response here.
- Apply the product rule:
add response here.
- Combine the outer function derivative:
add response here.
- Simplify:
add response here.