MATH 171 - Derivative Worksheet with 48 problems for practicing differentiation of functions.

Problem: Differentiate the given functions with respect to \( x \).

We will solve a few of these problems step by step to illustrate the process. The general techniques used include:

1. Power Rule: \(\frac{d}{dx} [x^n] = n x^{n-1}\)
2. Product Rule: \(\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\)
3. Quotient Rule: \(\frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\)
4. Chain Rule: \(\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)\)
5. Logarithmic Differentiation for complex expressions.

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#### Problem 1: \( f(x) = 4x^3 - 5x^4 \)

Using the power rule:
\[
f'(x) = \frac{d}{dx}[4x^3] - \frac{d}{dx}[5x^4]
\]
\[
f'(x) = 4 \cdot 3x^2 - 5 \cdot 4x^3
\]
\[
f'(x) = 12x^2 - 20x^3
\]

#### Problem 2: \( f(x) = e^x \sin x \)

Using the product rule:
\[
f'(x) = \frac{d}{dx}[e^x \sin x] = e^x \frac{d}{dx}[\sin x] + \sin x \frac{d}{dx}[e^x]
\]
\[
f'(x) = e^x \cos x + \sin x \cdot e^x
\]
\[
f'(x) = e^x (\cos x + \sin x)
\]

#### Problem 3: \( f(x) = (x^4 + 3x)^{-1} \)

Using the chain rule:
\[
f(x) = (x^4 + 3x)^{-1}
\]
Let \( u = x^4 + 3x \). Then \( f(u) = u^{-1} \).
\[
f'(x) = \frac{d}{dx}[u^{-1}] \cdot \frac{d}{dx}[u]
\]
\[
f'(x) = -u^{-2} \cdot (4x^3 + 3)
\]
Substitute back \( u = x^4 + 3x \):
\[
f'(x) = -\frac{4x^3 + 3}{(x^4 + 3x)^2}
\]

#### Problem 4: \( f(x) = 3x^2 (x^3 + 1)^7 \)

Using the product rule and chain rule:
\[
f(x) = 3x^2 \cdot (x^3 + 1)^7
\]
\[
f'(x) = \frac{d}{dx}[3x^2] \cdot (x^3 + 1)^7 + 3x^2 \cdot \frac{d}{dx}[(x^3 + 1)^7]
\]
\[
f'(x) = 6x \cdot (x^3 + 1)^7 + 3x^2 \cdot 7(x^3 + 1)^6 \cdot 3x^2
\]
\[
f'(x) = 6x (x^3 + 1)^7 + 63x^4 (x^3 + 1)^6
\]
Factor out common terms:
\[
f'(x) = 3x (x^3 + 1)^6 \left[ 2(x^3 + 1) + 21x^3 \right]
\]
\[
f'(x) = 3x (x^3 + 1)^6 (2x^3 + 2 + 21x^3)
\]
\[
f'(x) = 3x (x^3 + 1)^6 (23x^3 + 2)
\]

#### Problem 5: \( f(x) = \cos(e^x) - 2x^2 \)

Using the chain rule and power rule:
\[
f'(x) = \frac{d}{dx}[\cos(e^x)] - \frac{d}{dx}[2x^2]
\]
\[
f'(x) = -\sin(e^x) \cdot \frac{d}{dx}[e^x] - 4x
\]
\[
f'(x) = -\sin(e^x) \cdot e^x - 4x
\]
\[
f'(x) = -e^x \sin(e^x) - 4x
\]

#### Problem 6: \( f(x) = \frac{x}{1 + x^2} \)

Using the quotient rule:
\[
f(x) = \frac{x}{1 + x^2}
\]
\[
f'(x) = \frac{(1 + x^2) \frac{d}{dx}[x] - x \frac{d}{dx}[1 + x^2]}{(1 + x^2)^2}
\]
\[
f'(x) = \frac{(1 + x^2) \cdot 1 - x \cdot 2x}{(1 + x^2)^2}
\]
\[
f'(x) = \frac{1 + x^2 - 2x^2}{(1 + x^2)^2}
\]
\[
f'(x) = \frac{1 - x^2}{(1 + x^2)^2}
\]

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Final Answers for Selected Problems:

\[
\boxed{
\begin{aligned}
1. & \quad f'(x) = 12x^2 - 20x^3 \\
2. & \quad f'(x) = e^x (\cos x + \sin x) \\
3. & \quad f'(x) = -\frac{4x^3 + 3}{(x^4 + 3x)^2} \\
4. & \quad f'(x) = 3x (x^3 + 1)^6 (23x^3 + 2) \\
5. & \quad f'(x) = -e^x \sin(e^x) - 4x \\
6. & \quad f'(x) = \frac{1 - x^2}{(1 + x^2)^2}
\end{aligned}
}
\]

This approach can be extended to solve all the problems in the worksheet. Let me know if you need solutions for more specific problems!