Differentiation: sum, product, quotient, trigonometric, inverse trigonometric, exponential, logarithmic, hyperbolic, implicit, explicit, and parametric functions.
- The sum rule states that the derivative of the sum of two functions is equal to the sum of their derivatives.
- The product rule states that the derivative of the product of two functions is equal to the first function times the derivative of the second, plus the second function times the derivative of the first.
- The quotient rule states that the derivative of the quotient of two functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
- The chain rule, also known as function of function rule, is used to differentiate composite functions, or functions of functions, and states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
- The derivative of trigonometric functions depends on the specific trigonometric function, but can generally be found using basic trigonometric identities.
- The derivative of inverse trigonometric functions is found using differentiation by substitution and the inverse function theorem.
- The derivative of exponential functions is equal to the function itself, multiplied by the natural logarithm of the base e.
- The derivative of logarithmic functions is found using the logarithmic differentiation technique, which involves taking the natural logarithm of both sides of the equation and differentiating using the chain rule.
- The derivative of hyperbolic functions is similar to that of trigonometric functions and can be found using basic identities.
- The differentiation of implicit, explicit, and parametric functions involves different techniques, but typically involves applying the chain rule and solving for the derivative.
Important Formulas to Remember in these topic
Differentiation Formulas
- Sum of Functions: $$\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$$
- Product of Functions: $$\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x)$$
- Quotient of Functions: $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{g(x)^2}$$
- Function of Function: $$\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$$
- Trigonometric Functions: $$\frac{d}{dx}\sin(x) = \cos(x), \quad \frac{d}{dx}\cos(x) = -\sin(x), \quad \frac{d}{dx}\tan(x) = \sec^2(x)$$
- Inverse Trigonometric Functions: $$\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arccos(x) = -\frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arctan(x) = \frac{1}{1+x^2}$$
- Exponential Functions: $$\frac{d}{dx}e^x = e^x$$
- Logarithmic Functions: $$\frac{d}{dx}\ln(x) = \frac{1}{x}, \quad \frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}$$
- Hyperbolic Functions: $$\frac{d}{dx}\sinh(x) = \cosh(x), \quad \frac{d}{dx}\cosh(x) = \sinh(x), \quad \frac{d}{dx}\tanh(x) = \operatorname{sech}^2(x)$$
- Implicit Functions: $$\frac{dy}{dx} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}$$
- Explicit Functions: $$\frac{d}{dx}y = \frac{dy}{dx}$$
- Parametric Functions: $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$