Derivatives

A ready-to-use page about derivatives rules, properties and notable cases

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derivative

Introduction

Derivatives, in the realm of mathematics, are a fundamental concept in calculus that measure how a function changes as its input changes. Essentially, a derivative represents the rate of change or the slope of a function at any given point. This concept is crucial for understanding and modeling real-world phenomena where change occurs, such as in physics, engineering, economics, and various other fields. To introduce derivatives, one typically starts with the concept of a function, which is a relationship between two sets of numbers or variables. When we have a function f(x), the derivative of this function, denoted as f'(x) or df(x)/dx, quantifies how f(x) changes as x changes. One of the foundational ideas in calculus is the limit process, which is used to define the derivative rigorously.

Derivatives have numerous applications:

Slope of a Curve: They provide the slope of the tangent line to the curve at any point, which helps in understanding the behavior of the function.
Optimization: In engineering and other fields, derivatives are used to find maximum and minimum values of functions, aiding in optimization problems.
Motion: In physics, derivatives describe velocity and acceleration, which are rates of change of position and velocity, respectively.
Economics and Biology: In these fields, derivatives model growth rates, decay, and other dynamic changes.

Derivatives are further generalized in higher dimensions, leading to concepts like partial derivatives, which deal with functions of multiple variables, and are fundamental in multivariable calculus and differential equations.

Understanding derivatives opens the door to deeper exploration in mathematics and its applications, providing a powerful tool for analysis and problem-solving in diverse scientific and engineering disciplines.

Definition

The derivative of f(x) at a point x=a is defined as: $$f'(a) = lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$$

Derivatives

Derivation rules and derivative properties

Expression		Equivalent	Expression		Equivalent
$$\frac{\mathrm{d}(f(t)+g(t))}{\mathrm{d}t}$$	$$\Rightarrow$$	$$\frac{\mathrm{d}(f(t))}{\mathrm{d}t} + \frac{\mathrm{d}(g(t))}{\mathrm{d}t}$$	$$\frac{\mathrm{d}(a\cdot f(t))}{\mathrm{d}t}, \ a \ \in \mathbb{R}$$	$$\Rightarrow$$	$$a \cdot \frac{\mathrm{d}(f(t))}{\mathrm{d}t}$$
$$\frac{\mathrm{d}(f(t)\cdot g(t))}{\mathrm{d}t}$$	$$\Rightarrow$$	$$g(t)\frac{\mathrm{d}(f(t))}{\mathrm{d}t} + f(t)\frac{\mathrm{d}(g(t))}{\mathrm{d}t}$$	$$\frac{\mathrm{d}^2(f(t))}{\mathrm{d}t^2}$$	$$\Rightarrow$$	$$\frac{\mathrm{d}\left( \frac{\mathrm{d}f(t)}{\mathrm{d}t} \right) }{\mathrm{d}t} $$
$$\frac{\mathrm{d}(\frac{f(t)}{g(t)})}{\mathrm{d}t}$$	$$\Rightarrow$$	$$\frac{f(t)\frac{\mathrm{d}(g(t))}{\mathrm{d}t} - g(t)\frac{\mathrm{d}(f(t))}{\mathrm{d}t}}{g^2(t)}$$	$$\frac{\mathrm{d}(f(g(t)))}{\mathrm{d}t}$$	$$\Rightarrow$$	$$ \frac{\mathrm{d}(f(g(t)))}{\mathrm{d}t} \cdot \frac{\mathrm{d}(g(t))}{\mathrm{d}t}$$

Notable cases

Expression		Equivalent	Expression		Equivalent
$$\frac{\mathrm{d}a}{\mathrm{d}t}, \ a \ \in \mathbb{R}$$	$$\Rightarrow$$	$$0$$	$$\frac{\mathrm{d}at}{\mathrm{d}t}, \ a \ \in \mathbb{R}$$	$$\Rightarrow$$	$$a$$
$$\frac{\mathrm{d}at^n}{\mathrm{d}t}$$	$$\Rightarrow$$	$$a\cdot n \cdot t^{n-1}$$	$$\frac{\mathrm{d}\frac{1}{t}}{\mathrm{d}t}$$	$$\Rightarrow$$	$$-\frac{1}{t^2}$$
$$\frac{\mathrm{d}be^{at}}{\mathrm{d}t}$$	$$\Rightarrow$$	$$b \cdot a \cdot e^{at}$$	$$\frac{\mathrm{d}bc^{at}}{\mathrm{d}t}$$	$$\Rightarrow$$	$$b \cdot a \cdot ln(c) c^{at}$$
$$\frac{\mathrm{d}ln(at)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$\frac{1}{t}$$	$$\frac{\mathrm{d}log_b(at)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$log_b(e)\frac{1}{t}$$
$$\frac{\mathrm{d}sin(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$cos(t)$$	$$\frac{\mathrm{d}cos(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$-sin(t)$$
$$\frac{\mathrm{d}tan(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$sec^2(t)$$	$$\frac{\mathrm{d}sec(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$sec(t)tan(t)$$
$$\frac{\mathrm{d}cot(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$-cosec^2(t)$$	$$\frac{\mathrm{d}cosec(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$-cosec(t)cot(t)$$
$$\frac{\mathrm{d}sin^{-1}(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$\frac{1}{\sqrt{1-x^2}}$$	$$\frac{\mathrm{d}cos^{-1}(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$-\frac{1}{\sqrt{1-x^2}}$$
$$\frac{\mathrm{d}tan^{-1}(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$\frac{1}{1+x^2}$$	$$\frac{\mathrm{d}cosh(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$sinh(t)$$
$$\frac{\mathrm{d}sinh(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$cosh(t)$$	$$\frac{\mathrm{d}tanh(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$sech^2(t)$$
$$\frac{\mathrm{d}sech(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$-sech(t)tanh(t)$$	$$\frac{\mathrm{d}cosech(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$-cosech(t)coth(t)$$
$$\frac{\mathrm{d}coth(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$-cosech^2(t)$$	$$\frac{\mathrm{d}cosh^{-1}(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$\frac{1}{\sqrt{t^2-1}}$$
$$\frac{\mathrm{d}sinh^{-1}(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$\frac{1}{\sqrt{t^2+1}}$$	$$\frac{\mathrm{d}tanh^{-1}(t)}{\mathrm{d}t}$$	$$\Rightarrow$$	$$\frac{1}{1-t^2}$$