Math Calculators ▶ Derivative Calculator
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An online derivative calculator helps to find the derivative of the function with respect to a given variable and shows you the step-by-step differentiation. For better understanding, you can take a look at the given examples to differentiate the function. You can use this differential calculator to simplify first, second, third, or up to 5 derivatives.
No doubt, an online derivative solver is the best way to take a derivative at any point and even assists you to solve partial derivatives. Well, this context provides you with the derivative rules, how to find derivative (step-by-step), and by using a calculator.
In mathematics, the “derivative” measures the sensitivity to change of output value with respect to a change in the input value but in calculus, derivatives are central tools.
Example:
In the case of a moving object with respect to the time the derivative is the change in velocity in a certain time. In simple words, it measures how quickly a moving object changes its position when time advances. Therefore, the derivative is the “instantaneous rate of change”, in the dependent variable to that of the independent variable.
The process of finding a derivative is known as differentiation. Consequently, a Differentiation calculator will be a great help for the quick identification of derivatives.
Did You Know!
Many statisticians have defined derivatives simply by the following formula:
The derivative of a function f is represented by d/dx* f. “d” is denoting the derivative operator and x is the variable. The derivatives calculator let you find derivative without any cost and manual efforts. However, the derivative of the “derivative of a function” is known as the second derivative and can be calculated with the help of a second derivative calculator. whenever you have to handle up to 5 derivatives along with the implication of differentiation rules just give a try to a derivative finder to avoid the risk of errors.
There are certain rules that can be used to find out derivatives. These beneficial rules help you work out the derivatives. By following them you can add subtract and understand that how to take a derivative. Have a look down below to learn about them:
| Common Functions | Function | Derivative |
|---|---|---|
| Constant | c | 0 |
| Line | x | 1 |
| ax | a | |
| Square | x2 | 2x |
| Square Root | √x | (½)x-½ |
| Exponential | ex | ex |
| ax | ln(a) ax | |
| Logarithms | ln(x) | 1/x |
| loga(x) | 1 / (x ln(a)) | |
| Trigonometry (x is in radians) | sin(x) | cos(x) |
| cos(x) | −sin(x) | |
| tan(x) | sec2(x) | |
| Inverse Trigonometry | sin-1(x) | 1/√(1−x2) |
| cos-1(x) | −1/√(1−x2) | |
| tan-1(x) | 1/(1+x2) | |
| Rules | Function | Derivative |
|---|---|---|
| Multiplication by constant | cf | cf’ |
| Power Rule | xn | nxn−1 |
| Sum Rule | f + g | f’ + g’ |
| Difference Rule | f – g | f’ − g’ |
| Product Rule | fg | f g’ + f’ g |
| Quotient Rule | f/g | (f’ g − g’ f )/g2 |
| Reciprocal Rule | 1/f | −f’/f2 |
| Chain Rule (as “Composition of Functions”) |
f º g | (f’ º g) × g’ |
| Chain Rule (using ’ ) |
f(g(x)) | f’(g(x))g’(x) |
| Chain Rule (using \( \frac{dy}{dx}\)) |
\( \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}\) | |
Here we are going to helps you to solve derivative problems according to the above-mentioned differentiation rules. So, let’s start!
Example:
What is the derivative of \(cos (x)\)?
Apart from manual calculations, you can look at the above table to find the derivative of \(cos(x)\)
$$ \frac {d} {dx} cos (x) $$
We can write as:
$$ = -sin(x) $$
Hence
$$ cos(x)’ = – sin(x) $$
Example:
What is \(\frac {d} {dx} x^2\) ?
We use Power Rule, Where \(n = 2\):
$$ \frac {d} {dx} x^n = nx^{n-1}$$
After putting \( n = 2\) in power rule formula
$$ \frac {d} {dx} x^2 = 2x^{2-1}$$
$$ = 2x$$
\( \frac {2} {x} \) is also \( 2x^{-1} \)
$$\frac {d} {dx} 2x^{-1} = 2\frac {d} {dx} x^{-1}$$
$$= 2 (-1) x^{-1-1}$$
So;
$$= -2x^{-2}$$
$$=\frac {-2} {x^2}$$
Example:
What is \(\frac {d} {dx} 3x^4\) ?
$$\frac {d} {dx} 3x^4 $$
Taking from Power Rule
$$\frac {d} {dx} x^4 = 4x^{4-1} = 4x^3 $$
$$ \frac {d} {dx} 3x^4 = 3\frac {d} {dx} x^4 = 3 * 4x^3 = 12x^3$$
According to Sum Rule:
The derivative of \(x + y = x’ + y’\)
Example:
What is derivative of \(x^3 + 13 x^2\)?
We take each derivative separately after that add them.
$$x^3 + 13 x^2$$
By using power Rule
$$\frac {d} {dx} (x^3 = 13x^2) = \frac {d} {dx} x^3 + \frac {d} {dx} 13x^2$$
Hence
$$= 3x^{3-1} + 13 * 2x^{2-1} = 3x^2 + 26x$$
According to Difference Rule:
The derivative of \( x – y = x’ – y’\)
Example:
What is \(\frac {d} {dy} (y^2 – 3y^4)\)?
We take each derivative separately after that add them.
By using Power Rule
$$\frac {d} {dy} (y^2 – 3y^4) = \frac {d} {dy} y^2 – \frac {d} {dy} 3y^4$$
$$= 2y^{2-1} – 3 * 4y^{4-1}$$
Hence
$$= 2y – 12y^3 $$
Example:
What is \(\frac {d} {dx} (3x^3 + x^2 -7x)\) ?
By using the Power Rule
$$\frac {d} {dx} (3x^3 + x^2 -7x)$$
$$= \frac {d} {dx} 3x^3 + \frac {d} {dx} x^2 – \frac {d} {dx} 7x$$
$$= 3 * 3x^{2-1} + 2x^{2-1} – 7 * 1$$
Hence
$$= 9x^2 + 2x – 7$$
According to Product Rule:
The derivative of \(xy = xy’ + x’y\)
Example:
What is the derivative of \(sin(x)cos(x)\) ?
If we put values in Product Rule:
$$x = sin$$
$$y = cos$$
After reading above table:
$$\frac {d} {dz} (sin(z) cos(z))$$
$$= sin(z) \frac {d} {dz} cos(z) + cos(z) \frac {d} {dz} sin(z)$$
So
$$= sin(z) (- sin(z)) + cos(z) . cos(z)$$
$$= – sin^2 (z) + cos^2 (z)$$
According to Quotient Rule:
$$(\frac {x} {y} )’ = \frac {xy’ – x’y} {y^2}$$
Example:
What is the derivative of \( \frac {sin(z)} {z}\) ?
$$\frac {d} {dz} (\frac {sin(z)} {z})$$
$$= \frac {z \frac {d} {dz} (sin(z)) – sin(z) \frac {d} {dz} z} {z^2}$$
Hence
$$= \frac {zcos(z) – sin(z) } {z^2}$$
According to Reciprocal Rule:
The derivative of \(\frac {1} {w} = \frac {-fw’} {w^2}\)
Example:
What is \( \frac {d} {dw} (\frac {1} {w})\)?
$$\frac {1} {w}$$
By using \(f(w)= w\) , we can see that \(f’(w) = 1\)
According to Chain Rule:
The derivation of \(f(g(x)) = f ‘(g(x))g'(x)\)
Example:
What is \(\frac {d} {dx} (cos(x^3))\) ?
$$\frac {dy} {dx} = \frac {dy} {du} . \frac {du} {dx}$$
Differentiate each value:
$$\frac {d} {dx} (cos(x^3))$$
$$f(h) = cos(h)$$
The value of \(h(x)\)
$$h(x) = x^3 $$
$$f ‘(h) = -sin(x)$$
$$h ‘(x) = 3x^2$$
According to above table the derivative of \(cos(x)\)
$$\frac {d} {dx} (cos(x^3)) = -sin(h(x))(3x^2)$$
$$= – 3x^2 sin(x^3)$$
Similarly
$$\frac {d} {dx} (cos(x^3)) = \frac {d} {du} cos(u) \frac {d} {x} x^3$$
$$= -sin(u) 3x^2$$
Hence
$$= -3x^2 sin(x^3)$$
To calculate the derivative, you have to follow a simple step by step procedure:
Input:
Output:
First of all, you have to take the partial derivative of z with respect to x. However, very next you have to assume the derivative again, with respect to y. x should remain constant. now pay attention to the phenomena of the cross partial as a measure of in what way the slope changes, with the change in the y variable. For clarification, you can take assistance from the first derivative calculator by solving a derivative problem.
The second derivative measures the rate at which the first derivative changes. The second derivative will demonstrate the increase or decrease in the slope of the tangent line. Hence with the support of a double derivative calculator, the rate of change of the original function can be monitored.
The order of differentiation or derivative does not matter at all. You can first differentiate with respect to the second derivative and then with respect to the first derivative or vice versa. For convenience, you could use the free second derivative calculator that computes first, second, or up to 5 differentiation step-by-step.
Logarithmic differentiation can be used to express the form \(y = f(x)g(x)\), a variable to the power of a variable. You can’t apply The power rule and the exponential rule in such a situation. You can try a logarithmic differentiation calculator that helps to solve your logarithmic differentiation problems stepwise.
Whenever there will be a derivative of a function, you are going to end up with another function that will provide the slope of the original function. For the derivative of a function, there must be the same limit from left to right for it to be differentiable at that point.
This Derivative Calculator demonstrates a step-by-step help to find the derivatives and derivative of the function. It follows the different rules of differentiation and anyone can handle simple and complex derivative calculations with this derivative finder. It is a great help for academic and learning purposes and supports students as well as professionals equally. Additionally, this differential calculator can evaluate the derivatives at the given point, whenever needed.
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