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

1. Power Function:

• Function:

$$f(x)=x^n$$

• Derivative:

$$f^{\prime }(x)=\frac{d}{dx}(x^n)=n{x}^{n-1}$$

2. Exponential Function:

• Function:

$$f(x)=e^x$$

• Derivative:

$$f^{\prime }(x)=\frac{d}{dx}(e^x)=e^x$$

• Function:

$$f(x)=a^x$$

• Derivative:

$$f^{\prime }(x)=\frac{d}{dx}(a^x)=a^x\ln (a)$$

3. Natural Logarithm

• Function:

$$f(x)=ln(x)$$

• Derivative:

$$f^{\prime }(x)=\frac{d}{dx}(\ln (x))=\frac{1}{x}$$

4. Trigonometric Functions

• Function:

$$f(x)=\sin (x)$$

• Derivative:

$$f^{\prime }(x)=\frac{d}{dx}(\sin (x))=\cos (x)$$

• Function:

$$f(x)=\cos (x)$$

• Derivative:

$$f^{\prime }(x)=\frac{d}{dx}(\cos (x))=-\sin (x)$$

• Function:

$$f(x)=\tan (x)$$

• Derivative:

$$f^{\prime }(x)=\frac{d}{dx}(\tan (x))={\sec }^2(x)$$

5. Hyperbolic Functions

• Function:

$$f(x)=\sinh (x)$$

• Derivative:

$$f^{\prime }(x)=\frac{d}{dx}(\sinh (x))=\cosh (x)$$

• Function:

$$f(x)=\cosh (x)$$

• Derivative:

$$f^{\prime }(x)=\frac{d}{dx}(\cosh (x))=\sinh (x)$$

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 $\frac{d}{dx}(x^n)=n{x}^{n-1}$

• Applying the power rule:

$$f^{\prime }(x)=\frac{d}{dx}(5x^3)=15x^2$$

Exercise 2

Function:

$$g(x)=\sqrt{x}$$

Solution:

• Rewrite the function with a fractional exponent: $\sqrt{x}=x^{1/2}$.

• Apply the power rule:

$$g(x)=x^{1/2}$$

$$g^{\prime }(x)=\frac{1}{2}{x}^{1/2-1}=\frac{1}{2}{x}^{-1/2}=\frac{1}{2\sqrt{x}}$$

Exercise 3

Function:

$$h(x)=\ln (x)$$

Solution:

• The derivative of the natural logarithm function is $\frac{1}{x}$

$$h^{\prime }(x)=\frac{d}{dx}(\ln (x))=\frac{1}{x}$$

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: $(u\cdot v{)}^{{}^{\prime }}=u^{{}^{\prime }}\cdot v+u\cdot v^{{}^{\prime }}$

• Differentiate each function:

$$u^{{}^{\prime }}=\frac{d}{dx}(x^2)=2x$$

$$v^{{}^{\prime }}=\frac{d}{dx}(e^x)=e^x$$

• Combine the results:

$$g^{{}^{\prime }}=x^2\cdot e^x+e^x\cdot 2x=e^x(x^2+2x)$$

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:

$$h^{{}^{\prime }}(x)=2x\cos (x^2)$$

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:

$$u^{{}^{\prime }}=\frac{d}{dx}(\ln (x))=\frac{1}{x}$$

$$v^{{}^{\prime }}=\frac{d}{dx}(e^{2x})=2e^{2x}$$

• Apply the product rule:

$$\frac{d}{dx}(\ln (x)\cdot e^{2x})=(\frac{1}{x}\cdot e^{2x})+(\ln (x)\cdot 2e^{2x})$$

$$=e^{2x}(\frac{1}{x}+2\ln (x))$$

• Combine the outer function derivative:

$$f^{\prime }(x)=3{(\ln (x)\cdot e^{2x})}^2\cdot e^{2x}(\frac{1}{x}+2\ln (x))$$

• Simplify:

$$f^{\prime }(x)=3e^{4x}(\ln (x){)}^2(\frac{1}{x}+2\ln (x))$$