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Derivatives and Rules
Derivatives and Rules
The derivative measures instantaneous rate of change and slope of the tangent line.
Definition
$$f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$
Basic Rules
| Function | Derivative |
|---|---|
| $c$ | $0$ |
| $x^n$ | $nx^{n-1}$ |
| $cf(x)$ | $cf'(x)$ |
| $f+g$ | $f'+g'$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $e^x$ | $e^x$ |
Example 1
Differentiate $f(x)=3x^4-5x+8$.
$$f'(x)=12x^3-5$$
Example 2
Differentiate $g(x)=x^3+\sin x$.
$$g'(x)=3x^2+\cos x$$
Product and Quotient Rules
$$\frac{d}{dx}[uv]=u'v+uv'$$
$$\frac{d}{dx}\left[\frac{u}{v}\right]=\frac{u'v-uv'}{v^2}$$
Example 3
Differentiate $h(x)=x^2e^x$.
$$h'(x)=2xe^x+x^2e^x=e^x(x^2+2x)$$
Practice Problems
1. $\frac{d}{dx}(7x^5)$
2. $\frac{d}{dx}(x^2-4x+9)$
3. $\frac{d}{dx}(\sin x+\cos x)$
4. $\frac{d}{dx}(x^3e^x)$
5. $\frac{d}{dx}\left(\frac{x^2+1}{x}\right)$
6. What does $f'(a)=0$ often indicate?
Show Answer Key
1. $35x^4$
2. $2x-4$
3. $\cos x-\sin x$
4. $3x^2e^x+x^3e^x$
5. $1-\frac{1}{x^2}$
6. A critical point, possibly a local max or min.