M171 Calculus I · Unit 2: Computing derivatives · on Exam 2
§2.1 Elementary derivative rules
Study notes for Elementary derivative rules in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Constant, power, constant-multiple and sum/difference rules let you differentiate any polynomial term by term. The power rule works for any real exponent, so rewrite roots and reciprocals as powers first: $\sqrt{x}=x^{1/2}$, $\frac1{x^3} = x^{-3}$.
- Exponentials: $\frac{d}{dx}[e^x]=e^x$ and $\frac{d}{dx}[a^x] = a^x\ln a$. Note $e^x$ is its own derivative; $x^e$ uses the power rule.
- Derivatives of sums are sums of derivatives, but derivatives of products are not products of derivatives (see §2.3).
Key formulas
Power rule
$$\frac{d}{dx}\big[x^n\big] = n\,x^{n-1}$$
Constant / constant multiple
$$\frac{d}{dx}[c] = 0,\qquad \frac{d}{dx}[c\,f(x)] = c\,f'(x)$$
Sum / difference
$$\frac{d}{dx}[f\pm g] = f' \pm g'$$
Exponentials
$$\frac{d}{dx}[e^x] = e^x,\qquad \frac{d}{dx}[a^x] = a^x\ln a$$
Worked example
Differentiate $f(x) = 4x^5 - \dfrac{3}{x^2} + 6\sqrt{x} - 2^x + 7$.
Show the solution
Rewrite: $4x^5 - 3x^{-2} + 6x^{1/2} - 2^x + 7$. Then $f'(x) = 20x^4 + 6x^{-3} + 3x^{-1/2} - 2^x\ln 2$.
Common mistakes
- $\frac{d}{dx}[2^x] = x\,2^{x-1}$ is wrong; the base is constant, so the answer is $2^x\ln2$.
- Forgetting the derivative of a constant is $0$ (not the constant).
Exam tipRewrite before you differentiate. Every fraction with x in the denominator and every root becomes a power.
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