Differentiation and Equation Solving

A set of vocabulary-style practice flashcards covering basic and complex differentiation rules, trigonometric derivatives, quotient and chain rules, and algebraic properties/exponential equations.

Derivative of $F(x)=4$

$$F'(x)=0$$

Derivative of $F(x)=4x$

$$F'(x)=4$$

Derivative of $F(x)=4x^3$

$$F'(x)=12x^2$$

Derivative of $F(x)=4(3x^3)$

$$F'(x) = 4 \times (9x^2) = 36x^2$$

Derivative of $\ln(x)$

$$\frac{1}{x}$$

Derivative of $\log(x)$ ($\text{base 10}$)

$$\frac{1}{x \ln(10)}$$

Derivative of $\log_5(x)$

$$\frac{1}{x \ln(5)}$$

Derivative of $e^x$

$$e^x$$

Derivative of $\sin(x)$

$$\cos(x)$$

Derivative of $\cos(x)$

$$-\sin(x)$$

Derivative of $-\sin(x)$

$$-\cos(x)$$

Derivative of $-\cos(x)$

$$\sin(x)$$

Derivative of $\tan(x)$

$\frac{1}{\cos^2(x)}$ which is also equal to $1 + \tan^2(x)$

Derivative of $\sin(ax+b)$

$$a \cos(ax+b)$$

Derivative of $\cos(ax+b)$

$$-a \sin(ax+b)$$

Sum Rule ($S(x)=f(x) + g(x)$)

$$S'(x)= f'(x) + g'(x)$$

Product Rule ($P(x)=f(x) \times g(x)$)

$$P'(x)= f'(x) \times g(x) + f(x) + g'(x)$$

Quotient Rule ($Q(x) = \frac{f(x)}{g(x)}$)

$$Q'(x) = \frac{\text{nat-tan}}{n^2}$$

Chain Rule ($K(x)= f(g(x))$)

$$K'(x)=f'(g(x)) \cdot g'(x)$$

Derivative of $W(x)=\sqrt{2x^2 + 10}$

$$W'(x) = 2x \cdot (2x^2 + 10)^{-\frac{1}{2}} = \frac{2x}{\sqrt{2x^2+10}}$$

Derivative of $K(x)=(3x + 2)^5$

$$K'(x)=5 \times (3x+2)^4 \times 3 = 15(3x+2)^4$$

Derivative of $M(x)= 6^{3x+1}$

$$M'(x) = 3 \ln(6) \cdot 6^{3x+1}$$

Derivative of $L(x)=\log_5(3x^2 + 1)$

$$L'(x) = \frac{6x}{(3x^2+1) \ln(5)}$$

Derivative of $E(x)=e^{5x+2}$

$$E'(x)= 5e^{5x+2}$$

Solution property for $\frac{A}{B} = \frac{C}{B}$

$$A = C$$

Solution property for $\frac{A}{B} = \frac{C}{D}$

$$A \cdot D = B \cdot C$$

Solution property for $\frac{A}{B} = \frac{A}{D}$

$A = 0$ or $B = D$

Exponential negative identity ($e^{-1}$)

$$e^{-1} = \frac{1}{e}$$

Natural logarithm definition

\ln(x) = \log_e(x)$ houses the base e$$

Solution to $e^x = 8$

$$x = \ln(8)$ house

Solution to $3e^{2x+1} = 15$

$$x = -\frac{1}{2} + \frac{1}{2} \ln(5)$$

Solution to $4\ln(x) = 6$

$$x = e^{1.5}$$

Fractional exponent definition for $16^{1/2}$

$$\sqrt{16}$$

Exponential form of $\sqrt{x}$

$$x^{1/2}$$

Exponential form of $x \sqrt{x}$

$$x^{1.5}$$

Exponential form of $8^{1/3}$

$$\sqrt[3]{8}$$

Negative exponent values ($4^{-1}$ and $3^{-2}$)

$4^{-1} = \frac{1}{4}$ and $3^{-2} = \frac{1}{9}$

Logarithmic base identity for $\log_{1/3}(x)$

$$-\log_3(x)$$

Solution to $\log_2(x-2)=3$

$$x-2 = 2^3$$

Solution to $2^x = 8$

$$x = \log_2(8) = 3$$