MATH 1500: Differentiation Rules

What’s the constant rule?

d/dx (C) = 0

What’s the power rule?

d/dx (xⁿ) = nx^n-1

What’s the constant multiple rule?

d/dx [C(f(x))] = (C) d/dx f(x)

What’s the definition of a derivative?

$$f^{\prime}\left(x\right)=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$$

What’s the product rule?

\frac{d}{\differentialD x}\left\lbrack f\left(x\right)\cdot g\left(x\right)\right\rbrack=f^{\prime}\cdot g+f\cdot g^{\prime}

What’s the quotient rule?

\frac{d}{\differentialD x}\left(\frac{f}{g}\right)=\frac{gf^{\prime}-fg^{\prime}}{g^2}

What’s the chain rule?

\frac{d}{\differentialD x}\left\lbrack f\left(g\left(u\right)\right)\right\rbrack=f^{\prime}\left(g\left(u\right)\right)\cdot g^{\prime}\left(u\right)\cdot u^{\prime}

How do you differentiate exponential functions featuring euler’s number?

\frac{d}{\differentialD x}\left(e^{u}\right)=e^{u}\cdot u^{\prime}

lne = 1

How do you differentiate exponential functions featuring a constant?

\frac{d}{\differentialD x}\left\lbrack a^{u}\right\rbrack=a^{u}\cdot u^{^{\prime}}\cdot\ln a

How do you differentiate natural logarithms?

\frac{d}{\differentialD x}\left\lbrack\ln\left(u\right)\right\rbrack=\frac{u^{\prime}}{u}

How do you differentiate logarithms?

\frac{d}{\differentialD x}\left\lbrack\log_{a}\left(u\right)\right\rbrack=\frac{u^{\prime}}{u\cdot\ln a}

How do you differentiate stuff like x^x?

1. Set y = x^x

2. Move around lnx^x’s exponent

3. Differentiate both sides

4. Sub y for x^x

What is the reciprocal identity of sinθ?

$$\sin\theta=\frac{1}{\csc\theta}$$

What is the reciprocal identity of cosθ?

$$\cos\theta=\frac{1}{\sec\theta}$$

What is the reciprocal identity of tanθ?

$$\tan\theta=\frac{1}{\cot\theta}$$

What’s the quotient identity of tanθ?

$$\tan\theta=\frac{\sin\theta}{\cos\theta}$$

What’s the quotient identity of cotθ?

$$\cot\theta=\frac{\cos\theta}{\sin\theta}$$

What’s the pythagorean identity of sin²θ?

$$\sin^2\theta=1-\cos^2\theta$$

What’s the pythagorean identity of sec²θ?

$$\sec^2\theta=1+\tan^2\theta$$

What’s the pythagorean identity of csc²θ?

$$\csc^2\theta=1+\cot^2\theta$$