`sin(sqrt(e^x + a)/2)`

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Our calculator will calculate the derivative of any function using the common rules of differentiation (product rule, quotient rule, chain rule, etc.). It can show the steps and interactive graphing for both input and result function. It can handle polynomial, rational, irrational, exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic and inverse hyperbolic functions. Also, it will evaluate the derivative at the given point, if needed. You can pick a random function using the 'Random function' button. You can also choose the order of the derivative by entering a value from min 1 to max 5 in 'Times?'.

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What is a derivative of a function?

A derivative of a function refers to the expression that, generally, represents the slope of a function at any point within its domain.

If a function is defined as , then the derivative of the function is denoted by or or simply .

For instance; if a straight line passes through points (1,0) and (2,4), then its derivative which is the slope of the line is

When we find the derivative of a function, we say, we are differentiating the function. If a function allows, we can repeatedly find the derivative of the derivatives. The first derivative is usually termed as slope or gradient or simply derivative. However, if we are interested in other derivatives other than the first one, it is always specified as second, third, or fourth derivatives, and so on.

The second derivative implies differentiating the function twice or simply, the derivative of the first derivative. We denote it as

The third derivative is the derivative of the second derivative. It is also similar to differentiating the function thrice. We have

Significance of derivative

The first derivative represents the rate of change of one quantity with respect to another. The first derivative is used in finance to quantify the rate of investment, the rate of growth of an investment, the rate of inflation, the rate of growth on production among others. It is also used in mechanics to quantify the speed, velocity, and rate of change of momentum among others. It is also used to measure the rate of change of energy. In the chemical field, it is used to measure the rate of change of chemical concentrations, the rate of reaction among

In mechanics: Velocity
Velocity or speed is the rate of change of displacement of distance respectively. It is used to measure the rate at which a body moves.

Example:
A train covered a distance of 186 miles in two hours. Find the average speed of the train.

In financial matters: Rate of inflation:
The rate of inflation is a measure of how the price increases or decreases on the market over a given period. It is expressed as a percentage.

Example:
The price of one gallon of gas was $3.034 a year ago. If the current price is 3.190, find the inflation rate during that year.
The inflation is

In financial matters: Rate of income growth:
The rate of income growth in a country can be measured in terms of the derivative, or gradient of the average income of the person in that country. This can be measured using per-capita income.

Example:
The per-capita income of the US was in 1990 and in 2000. Determine the rate of income growth over the 10 years.

Types of derivative

The derivatives are categorized into two main types, ordinary derivative or partial derivative. However, we use the term derivative to simply imply the first ordinary derivative of a function. When we have an expression that is a function of one variable only, then the derivative is an ordinary derivative. For instance, we have y = f(x) or h = g(t), then the first derivatives will simply be total and will be given as

Example:

The derivative of with respect to x is

If the function has more than one variable, then we can find the derivative with respect to one variable as we make another or others constant. This type of derivative is said to be partial. For instance, when the function is y = f(t,s) where t and s are other variables, then

i. The partial derivative of y with respect to t is

ii. The partial derivative of y with respect to s is

Example:

The derivative of with respect to x and y is

and

If the function has three variables, such as f = f(r, θ, z), then we have

i. The partial derivative of y with respect to r is

ii. The partial derivative of y with respect to θ is

iii. The partial derivative of y with respect to z is

Example:

The derivative of with respect to x, y and z is



Basic rules of derivative
In this section, we consider the derivative rules for ordinary derivatives which also apply for partial derivatives.









Derivatives of power functions and polynomials.

Power functions are generally written as where k and n are constants while x is a variable. The derivative of the function is

Example
The derivative of is
The polynomial is composed of terms that are power functions, therefore, its derivative is determined by differentiating term by term. If the term is a constant, for instance, h(x) = k where is a constant, then h'(x) = 0 because

and

Example
The derivative of is

Product rule

If a function f(x) is a product of two functions, g(x) and h(x) that are differentiable, then the derivative of f(x) = g(x) * h(x) is

Also written as

In simple terms, if

Quotient rule

If a function f(x) is a quotient of two functions, g(x) and h(x) that are differentiable, then the derivative of is is

In simple terms, if

Power rule

If a function f(x) is expressed as, then its derivative is

Example
The derivative of is

Chain rule

The chain rule is a method for differentiation that works for composite functions. It can also be used to differentiate a power function because such function can be expressed as a composite function. If f(x) = g(h(x)) then f=g(h) and h=h(x) hence

Example
Consider . Let then so that and

Derivative of trigonometric functions

We provide a table showing the derivatives of trigonometric functions

Example:
Find the derivative of
We use the chain rule: Let then . We have h'(x) = 4x and f'(h) = cos h
Therefore,

Example:
Find the derivative of
Let $$h(x)=\sec (4 x)$$ and $$f(h)=\sqrt{h}=h^{\frac{1}{2}}$$. We have by chain rule $$h^{\prime}(x)=4 \sec (4 x) \tan (4 x)$$ and $$ f^{\prime}(h)=\frac{1}{2} h^{\frac{1}{2}-1}=\frac{1}{2} h^{-\frac{1}{2}}=\frac{1}{2 \sqrt{h}} $$ Thus, $$ \frac{d f}{d x}=\frac{d f}{d h} \cdot \frac{d h}{d x}=\frac{1}{2 \sqrt{h}} \cdot 4 \sec (4 x) \tan (4 x)=\frac{4 \sec (4 x) \tan (4 x)}{2 \sqrt{\sec (4 x)}}=2 \sqrt{\sec (4 x)} \tan (4 x) $$

Derivative of exponential functions

The derivative of function is

If a = e = 2.71..., then a = ln e = 1, hence we have derivative of being

Example
Find the derivative of
In this case, . To get , we use the product rule. $$ g^{\prime}(x)=2 x(\tan x)+\left(x^{2}\right) \sec ^{2} x=2 x \tan x+x^{2} \sec ^{2} x $$ Therefore, $$ f^{\prime}(x)=e^{g(x)} \cdot g^{\prime}(x)=\left(2 x \tan x+x^{2} \sec ^{2} x\right) e^{x^{2} \tan x} $$

Example
Find the derivative of
Let $$g(x)=\frac{x}{x^{2}+1}$$, using the quotient rule, $$ g^{\prime}(x)=\frac{1\left(x^{2}+1\right)-2 x(x)}{\left(x^{2}+1\right)^{2}}=\frac{x^{2}+1-2 x^{2}}{\left(x^{2}+1\right)^{2}}=\frac{1-x^{2}}{\left(x^{2}+1\right)^{2}} $$ Therefore, the derivative is $$ f^{\prime}(x)=a^{g(x)} \cdot g^{\prime}(x) \ln a=2^{\frac{x}{x^{2}+1}} \cdot \frac{1-x^{2}}{\left(x^{2}+1\right)^{2}} \cdot \ln 2=\frac{\left(1-x^{2}\right) 2^{\frac{x}{x^{2}+1}} \ln 2}{\left(x^{2}+1\right)^{2}} $$

Derivative of logarithmic functions

The derivative of the function is If a=e=2.71..., then ln a=ln e=1, hence we have the derivative of being

Example
Find the derivative of
Let , then .Hence, and
Therefore,

Example
Find the derivative of
Let $$g(x)=\cos (2 x)$$, then $$f(g)=\log _{3} g$$. By the chain rule, we have, $$g^{\prime}(x)=-2 \sin (2 x)$$ and $$f^{\prime}(g)=\frac{1}{g \ln 3}$$ Therefore, $$ f^{\prime}(x)=\frac{g^{\prime}(x)}{g(x) \ln a}=\frac{1}{g \ln 3} \cdot(2 \cos (2 x))=\frac{2 \sin (2 x)}{\cos (2 x) \ln 3}=\frac{2 \tan (2 x)}{\ln 3} $$

Constant Multiple Rule
Sum (or Difference) Rule
Power Rule
Product Rule
Quotient Rule
Chain Rule
Exponential Functions
Natural Logarithm
sine function
cosine function