Calculus Strategy Guide

The Power Rule

The most fundamental rule of differentiation. If you have a variable raised to a constant power, multiply the coefficient by the power, then reduce the exponent by 1.

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

Examples:

• \( \frac{d}{dx} 5x^3 = 15x^2 \)
• \( \frac{d}{dx} x^{-2} = -2x^{-3} \)
• \( \frac{d}{dx} \sqrt{x} = \frac{d}{dx} x^{1/2} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} \)
• \( \frac{d}{dx} 7 = 0 \) — the derivative of any constant is zero.

Key insight: The power rule works for any real exponent — fractions, negatives, decimals. Always rewrite roots and fractions as exponents first.

Sum & Difference Rule

Derivatives distribute across addition and subtraction. Differentiate each term individually.

\( \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) \)

Example: \( \frac{d}{dx}(4x^3 - 7x + 2) = 12x^2 - 7 \)

The Chain Rule

Used for composite functions — a function inside another function. Take the derivative of the outer function (leaving the inner function untouched), then multiply by the derivative of the inner function.

\( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \)

Examples:

• \( \frac{d}{dx}(3x+2)^5 = 5(3x+2)^4 \cdot 3 = 15(3x+2)^4 \)
• \( \frac{d}{dx}\sin(x^2) = \cos(x^2) \cdot 2x = 2x\cos(x^2) \)
• \( \frac{d}{dx}e^{4x} = e^{4x} \cdot 4 = 4e^{4x} \)
• \( \frac{d}{dx}\ln(3x) = \frac{1}{3x} \cdot 3 = \frac{1}{x} \)

Mental model: Peel the onion layer by layer — outer derivative first, always multiply by the inner derivative at the end.

The Product Rule

When two functions are multiplied together, you cannot just multiply their derivatives. Use:

\( \frac{d}{dx}[f \cdot g] = f'g + fg' \)

Memory trick: "First times derivative of second, plus second times derivative of first."

Example: \( \frac{d}{dx}[x^2 \sin x] = 2x\sin x + x^2 \cos x \)

The Quotient Rule

For a function divided by another function:

\( \frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2} \)

Memory trick: "Low D-High minus High D-Low, over Low squared, watch it go!"

Example: \( \frac{d}{dx}\left[\frac{x^2}{x+1}\right] = \frac{2x(x+1) - x^2(1)}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2} \)

Pro-tip: When the denominator is a monomial (like \(x^2\)), it's often faster to split the fraction and use the power rule instead.

The Six Trig Derivatives

You must memorize all six. Notice the pattern: co-functions (cosine, cotangent, cosecant) have a negative sign.

With Chain Rule: \( \frac{d}{dx}\sin(g(x)) = \cos(g(x)) \cdot g'(x) \)

Example: \( \frac{d}{dx}\cos(5x^2) = -\sin(5x^2) \cdot 10x = -10x\sin(5x^2) \)

Inverse Trig Derivatives

These come up frequently in integration too (as antiderivatives).

Exponentials

\(e^x\) is unique — it is its own derivative, the only function for which this is true.

Example: \( \frac{d}{dx} 2^x = 2^x \ln 2 \approx 0.693 \cdot 2^x \)

Logarithms

Example: \( \frac{d}{dx}\ln(x^2 + 1) = \frac{2x}{x^2+1} \)

Logarithmic Differentiation: For functions like \(y = x^x\), take \(\ln\) of both sides first, differentiate implicitly, then solve for \(y'\).

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The Reverse Power Rule

The antiderivative of \(x^n\) is found by adding 1 to the exponent and dividing by the new exponent. Always add \(+ C\) (the constant of integration).

\( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \)

Examples:

• \( \int 6x^2 \, dx = 2x^3 + C \)
• \( \int \sqrt{x} \, dx = \int x^{1/2}\,dx = \frac{2}{3}x^{3/2} + C \)
• \( \int \frac{1}{x^2}\,dx = \int x^{-2}\,dx = -x^{-1} + C = -\frac{1}{x} + C \)

Exception: \( \int \frac{1}{x}\,dx = \ln|x| + C \) — the \(n=-1\) case that breaks the power rule formula.

U-Substitution

The reverse of the chain rule. Use it when you spot a composite function and its inner function's derivative is also present in the integrand.

Steps:

1. Choose \(u\) = the inner function.
2. Compute \(du = u'\,dx\) and solve for \(dx\).
3. Substitute \(u\) and \(dx\) into the integral — it should simplify cleanly.
4. Integrate with respect to \(u\).
5. Back-substitute the original expression for \(u\).

Example: \( \int 2x\cos(x^2)\,dx \)

Let \(u = x^2\), so \(du = 2x\,dx\).

\( \int \cos(u)\,du = \sin(u) + C = \sin(x^2) + C \)

For definite integrals: change the limits of integration to \(u\)-values so you don't need to back-substitute.

Integration By Parts

The reverse of the product rule. Use when integrating a product of two unrelated function types (polynomial × trig, polynomial × exponential, etc.).

\( \int u\,dv = uv - \int v\,du \)

Choosing u — LIATE Priority:

1. Logarithmic: \(\ln x\), \(\log_a x\)
2. Inverse Trig: \(\arctan x\), \(\arcsin x\)
3. Algebraic: \(x^n\), polynomials
4. Trigonometric: \(\sin x\), \(\cos x\)
5. Exponential: \(e^x\), \(a^x\)

Pick \(u\) from the highest priority item present, and \(dv\) is everything else.

Example: \( \int x e^x\,dx \)

Let \(u = x\) (Algebraic), \(dv = e^x\,dx\) (Exponential).
Then \(du = dx\), \(v = e^x\).
\( \int xe^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C = e^x(x-1)+C \)

Note: Sometimes you need to apply integration by parts twice for problems like \(\int x^2 e^x\,dx\).

Example 1 — Chain Rule + Power Rule

Find \( \frac{d}{dx}(x^2 + 5)^4 \)

Step 1: Identify outer function: \((\ \cdot\ )^4\) and inner function: \(x^2 + 5\).

Step 2: Outer derivative: \(4(\cdot)^3\), leave inner alone → \(4(x^2+5)^3\).

Step 3: Multiply by inner derivative: \(\frac{d}{dx}(x^2+5) = 2x\).

Answer: \(4(x^2+5)^3 \cdot 2x = \boxed{8x(x^2+5)^3}\)

Example 2 — Product Rule + Chain Rule

Find \( \frac{d}{dx}[x^3 \sin(2x)] \)

Step 1: Product rule: \(f = x^3\), \(g = \sin(2x)\).

Step 2: \(f' = 3x^2\); \(g' = \cos(2x) \cdot 2 = 2\cos(2x)\) (chain rule).

Answer: \(3x^2\sin(2x) + 2x^3\cos(2x)\)

Example 3 — U-Substitution

Find \( \int \frac{x}{x^2+1}\,dx \)

Step 1: Let \(u = x^2 + 1\), so \(du = 2x\,dx\), meaning \(x\,dx = \frac{du}{2}\).

Step 2: Substitute: \(\int \frac{1}{u} \cdot \frac{du}{2} = \frac{1}{2}\int \frac{du}{u} = \frac{1}{2}\ln|u| + C\).

Answer: \(\boxed{\frac{1}{2}\ln(x^2+1) + C}\)

Example 4 — Integration By Parts

Find \( \int x\ln x\,dx \)

Step 1: LIATE → \(u = \ln x\) (Logarithmic), \(dv = x\,dx\) (Algebraic).

Step 2: \(du = \frac{1}{x}\,dx\), \(v = \frac{x^2}{2}\).

Step 3: \(\int x\ln x\,dx = \frac{x^2}{2}\ln x - \int \frac{x^2}{2} \cdot \frac{1}{x}\,dx = \frac{x^2}{2}\ln x - \frac{1}{2}\int x\,dx\).

Answer: \(\boxed{\frac{x^2}{2}\ln x - \frac{x^2}{4} + C}\)

• Forgetting the chain rule: \(\frac{d}{dx}\sin(3x) \neq \cos(3x)\). You must multiply by \(3\): \(3\cos(3x)\).
• Applying the power rule to exponentials: \(\frac{d}{dx}e^x \neq xe^{x-1}\). The power rule is for \(x^n\), not \(a^x\).
• Distributing derivatives incorrectly: \(\frac{d}{dx}(f \cdot g) \neq f' \cdot g'\). Use the product rule.
• Forgetting \(+C\): Indefinite integrals always need a constant of integration. Losing points on this is avoidable.
• Wrong sign on trig derivatives: It's \(\frac{d}{dx}\cos x = -\sin x\), not \(+\sin x\). Co-functions have negatives.
• Skipping parentheses in chain rule: When differentiating \(\sin(3x^2)\), write \(\cos(3x^2)\) not \(\cos \cdot 3x^2\). Group clearly.
• Treating \(\ln\) as \(\log_{10}\): In calculus, \(\ln\) means natural log (base \(e\)). They are different functions.
• Forgetting absolute value in \(\ln|x|+C\): The antiderivative of \(\frac{1}{x}\) is \(\ln|x|+C\), not \(\ln x + C\).

How to Actually Get Good at Derivatives

1. Memorize the table, don't derive it. On an exam, you don't have time to re-derive \(\frac{d}{dx}\sin x = \cos x\). Drill the table until it's automatic.
2. Identify the rule before you write anything. Look at the function structure: Is it composite? (Chain Rule) A product? (Product Rule) A quotient? (Quotient Rule or rewrite). Rushing leads to using the wrong rule.
3. Rewrite first, differentiate second. Simplify \(\frac{x^3+x}{x}\) to \(x^2+1\) before differentiating. Simplify \(\sqrt[3]{x^2}\) to \(x^{2/3}\). Fewer rules → fewer mistakes.
4. Practice daily, not in bulk. 15 minutes of practice every day beats 3 hours the night before. This is exactly why Differdle gives daily problems.
5. Check your work by differentiating your integral. If you found an antiderivative, differentiate it — you should get the original integrand back.
6. Make a mistake card. Every time you make an error on a practice problem, write it down. Review these before exams — you're far more likely to make the same mistake again than a new one.