Calculus Derivatives and Trigonometric Identities

Flashcards reviewing essential differentiation rules for algebraic, exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions, along with core trigonometric identities.

Derivative of $x^n$

$$n \cdot x^{n-1}$$

Derivative of $e^x$

$$e^x$$

Derivative of $b^x$

$$b^x \cdot \ln(b)$$

Derivative of $\ln(x)$

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

Derivative of $\log_b(x)$

$$\frac{1}{x \cdot \ln(b)}$$

Derivative of $\sin(x)$

$$\cos(x)$$

Derivative of $\cos(x)$

$$-\sin(x)$$

Derivative of $\tan(x)$

$$\sec^2(x)$$

Derivative of $\sec(x)$

$$\sec(x)\tan(x)$$

Derivative of $\csc(x)$

$$-\csc(x)\cot(x)$$

Derivative of $\cot(x)$

$$-\csc^2(x)$$

Derivative of $\sin^{-1}(x)$

$$\frac{1}{\sqrt{1-x^2}}$$

Derivative of $\cos^{-1}(x)$

$$-\frac{1}{\sqrt{1-x^2}}$$

Derivative of $\tan^{-1}(x)$

$$\frac{1}{1+x^2}$$

Derivative of $\sec^{-1}(x)$

$$\frac{1}{|x|\sqrt{1-x^2}}$$

Derivative of $\csc^{-1}(x)$

$$-\frac{1}{|x|\sqrt{1-x^2}}$$

Derivative of $\cot^{-1}(x)$

$$-\frac{1}{1+x^2}$$

Definition of $\sinh(x)$

$$\frac{e^x - e^{-x}}{2}$$

Definition of $\cosh(x)$

$$\frac{e^x + e^{-x}}{2}$$

$$\sinh(-x)$$

$$-\sinh(x)$$

$$\cosh(x)$$

$$\cosh(-x) = \cosh(x)$$

Fundamental Hyperbolic Identity

$$\cosh^2(x) - \sinh^2(x) = 1$$

Derivative of $\sinh(x)$

$$\cosh(x)$$

Derivative of $\cosh(x)$

$$\sinh(x)$$

Derivative of $\tanh(x)$

$$\text{sech}^2(x)$$

Derivative of $\text{csch}(x)$

$$-\text{csch}(x)\coth(x)$$

Derivative of $\text{sech}(x)$

$$-\text{sech}(x)\tanh(x)$$

Derivative of $\coth(x)$

$$-\text{csch}^2(x)$$

Derivative of $\sinh^{-1}(x)$

$$\frac{1}{\sqrt{1+x^2}}$$

Derivative of $\cosh^{-1}(x)$

$$\frac{1}{\sqrt{1-x^2}}$$

Derivative of $\tanh^{-1}(x)$

$$\frac{1}{1-x^2}$$

Derivative of $\text{csch}^{-1}(x)$

$$-\frac{1}{|x|\sqrt{x^2+1}}$$

Derivative of $\text{sech}^{-1}(x)$

$$-\frac{1}{x\sqrt{1-x^2}}$$

Derivative of $\coth^{-1}(x)$

$$\frac{1}{1-x^2}$$

Pythagorean Trigonometric Identity

$$\cos^2(\theta) + \sin^2(\theta) = 1$$

Cosine Sum and Difference Identity

$$\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$$

Sine Double-Angle Identity

$$\sin(2A) = 2\sin(A)\cos(A)$$

Cosine Double-Angle Identity

$$\cos(2A) = \cos^2(A) - \sin^2(A) = 1 - 2\sin^2(A) = 2\cos^2(A) - 1$$

Sine Power-Reduction Identity

$$\sin^2(x) = \frac{1-\cos(2x)}{2}$$

Cosine Power-Reduction Identity

$$\cos^2(x) = \frac{1+\cos(2x)}{2}$$

Sine Sum and Difference Identity

$$\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$$