AE 13: Derivation
Suggested answers
In this exercise, we will:
Practice 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:
\[ f'(x) = \frac{d}{dx}(5x^3) = 15x^2 \]
Exercise 2
Function:
\[ g(x) = \sqrt{x} \]
Solution:
Rewrite the function with a fractional exponent: \(\sqrt{x}=x^{1/2}\).
Apply the power rule:
\[ g(x)=x^{1/2} \]
\[ g'(x) = \frac{1}{2}x^{1/2-1} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} \]
Exercise 3
Function:
\[ h(x)=\ln(x) \]
Solution:
- The derivative of the natural logarithm function is \(\frac{1}{x}\)
\[ h'(x) = \frac{d}{dx}(\ln(x)) = \frac{1}{x} \]
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: \(u^4\)
Inner function: \(u=2x^3+3x\)
Apply the chain rule:
\[ \frac{d}{dx}(u^4)=4u^3 \cdot \frac{d}{dx}(u) \]
- Differentiate the inner function:
\[ \frac{d}{dx}(2x^3+3x)=6x^2+3 \]
- Combine the results:
\[ 4(2x^3 + 3x)^3 \cdot (6x^2 + 3) \]
Exercise 5
Function:
\[ \frac{d}{dx}(x^2 e^x) \]
Solution:
Identify the product of two functions: \(u=x^2\) and \(v=e^x\)
Apply the product rule: \((u \cdot v)^{'}=u^{'} \cdot v + u \cdot v^{'}\)
Differentiate each function:
\[ u^{'}=\frac{d}{dx}(x^2)=2x \]
\[ v^{'}=\frac{d}{dx}(e^x)=e^x \]
- Combine the results:
\[ g^{'}=x^2 \cdot e^x + e^x \cdot 2x = e^x(x^2+2x) \]
Exercise 6
Function:
\[ h(x)=\sin (x^2) \]
Solution:
Identify the outer function and inner function:
Outer function: \(\sin (u)\)
Inner function: \(u = x^2\)
Apply the chain rule: \(\frac{d}{dx}(\sin (u))=\cos (u) \cdot \frac{d}{dx}(u)\)
Differentiate the inner function:
\[ \frac{d}{dx}(x^2)=2x \]
- Combine the results:
\[ h^{'}(x)=2x\cos (x^2) \]
Advanced Derivatives
Exercise 7
Function:
\[ f(x)=(\ln (x) \cdot e^{2x})^3 \]
Solution
Identify the outer function and inner function
Outer function: \(u^3\)
Inner function: \(u=\ln (x) \cdot e^{2x}\)
Apply the chain rule: \(\frac{d}{dx}(u^3)=3u^2 \cdot \frac{d}{dx}(u)\)
Use the product rule to differentiate the inner function:
\[ u=\ln (x), v=e^{2x} \]
- Differentiate each function:
\[ u^{'}=\frac{d}{dx}(\ln (x)) = \frac{1}{x} \]
\[ v^{'}=\frac{d}{dx}(e^{2x}) = 2e^{2x} \]
- Apply the product rule:
\[ \frac{d}{dx}(\ln(x) \cdot e^{2x}) = \left( \frac{1}{x} \cdot e^{2x} \right) + \left( \ln(x) \cdot 2e^{2x} \right) \]
\[ = e^{2x} \left( \frac{1}{x} + 2 \ln(x) \right) \]
- Combine the outer function derivative:
\[ f'(x) = 3 \left( \ln(x) \cdot e^{2x} \right)^2 \cdot e^{2x} \left( \frac{1}{x} + 2 \ln(x) \right) \]
- Simplify:
\[ f'(x) = 3e^{4x} (\ln(x))^2 \left( \frac{1}{x} + 2 \ln(x) \right) \]