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derivative Calculator

Second Derivative Calculator

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An online second derivative calculator helps you to determine the second-order derivative for the entered equation. This calculator provides comprehensive calculations in a fraction of a second. Here you can get deep information about how to find second derivative of a given expression with power and chain rule.

What is Second Derivative?

In mathematics, the second derivative is known as the second-order derivative of the given expression. The finding process of the derivative of an equation is called the differentiation. Therefore, the process of determining the second-order derivative is called second-order differentiation. If the function is differentiated two times, then we can get the second-order derivative of a certain expression.

In simple words, the second derivative calculates how the rate of change of a certain quantity is changing itself. For example, find the second derivative of the position of an item with respect to the rate at which the velocity of the object is changing with respect to time (t) is:

$$ m = d(v) / d(t) = d^2 x / dt^2 $$

Where,

m = acceleration

t = time

x = position

d = instantaneous change

v = velocity

However, An Online Derivative Calculator helps to find the derivative of the function with respect to a given variable 

Power Rule for Second Derivative:

When apply the power rule twice, it will create the second derivative power rule that is used by double derivative calculator as:

$$ d^2 / dx^2 [x^n] = d / dx . d / dx [x^n] = d / dx [nx^{x-1} ] = n (n – 1) x^{n-2} $$

Notation for Second Derivation:

The second derivative of a function f(x) is normally denoted as:

F’’ = (f’)’

When using notation for derivatives, the second derivative of a dependent variable (y) with respect the independent variable (x) is written as \( d^2y / dx^2 \)

This is derived from

$$ d^2y / dx^2 = d / dx (dy / dx) $$

How to Calculate Second Derivative?

Calculating the second derivative of any expression has become handy if you have good knowledge about power and product rules.

Example:

Find the second derivative for \( d^2 / dx^2 sin (x) cos^3 (x) \).

Solution:

Given that:

$$ d^2 / dx^2 sin (x) cos^3 (x) $$

The second derivative calculator apply the product rule first:

$$ d / dx f(x) g(x) = f(x) d / dx g(x) + g(x) d / dx f(x) $$

$$ f(x)=cos^3(x); \text { to find } d / dx f(x): $$

$$ Let u=cos(x).$$

2nd derivative test calculator applies the power rule:

$$ U^3 \text{ goes to } 3u^2 $$

Then, second order derivative calculaor apply the chain rule. Multiply by d / dx cos(x):

The derivative of cosine is negative sine:

$$ d / dx cos(x) = −sin (x) $$

The result of the chain rule is:

$$ −3 sin (x) cos^2 (x) $$

$$ g(x) = sin (x); to find d / dx g(x): $$

The derivative of sine is cosine:

$$ d / dx sin (x) = cos (x) $$

The result is: \( −3sin^2 (x) cos^2(x) + cos^4(x) \)

Now, double derivative calculator simplify these obtained results:

$$ Cos (2x)^2 + cos (4x)^2 $$

Therefore, differentiate \( −3sin^2(x) cos^2(x) + cos^4(x) \) term by term:

Let u = cos (x)

Now, the second derivative test calculator applies the power rule: \( u^4 \text{ goes to } 4u^3 \)

Then, apply the chain rule. Multiply by \( d / dxcos(x) \):

The derivative of cosine is negative sine:

$$ d / dx cos(x) = −sin(x) $$

The result of the chain rule is:

$$ −4sin (x) cos^3 (x) $$

Now, second derivative calculator apply the product rule again for finding the second derivative:

$$ d / dx f(x) g(x) = f(x) d / dx g (x) + g(x) d / dx f(x) $$

$$ f(x) = cos^2 (x); \text { to find } d / dx f(x): $$

Let u = cos(x).

Apply the power rule: \( 9 u^2 \text{ goes to } 2u \)

Then, apply the chain rule. Multiply by d / dx cos (x):

The derivative of cosine is negative sine:

$$ d / dx cos (x) = −sin (x) $$

The result of the chain rule is:

$$ −2 sin (x) cos (x) $$

$$ g(x) = sin^2(x); \text{ to find} d / dx g(x): $$

Let u = sin (x).

Apply the power rule: u^2 \text{ goes to} 2u

Then, second order derivative calculator applies the chain rule. Multiply by d / dx sin (x):

The derivative of sine is cosine:

$$ d / dx sin (x) = cos(x) $$

The result of the chain rule is:

$$ 2 sin(x) cos(x) $$

So, the result is:

$$ −2sin^3 (x) cos (x) + 2sin (x) cos^3 (x) $$

Now, simplify :

$$ 6sin^3 (x) cos (x) − 6sin (x) cos^3 (x) $$

Hence,

$$ 6sin^3(x) cos(x) – 10 sin (x) cos^3 (x) $$

After simplifying the answer is:

$$ −sin (2x) − 2sin (4x) $$

However, an Online Integral Calculator helps you to evaluate the integrals of the functions with respect to the variable involved 

How Second Derivative Used For the Monotony Table?

The second derivative is usually used to know the slope variation of the curve that representing the function. For an interval:

  • a positive second derivative means an increment in the slope called convex function
  • a negative second derivative means a decrease in slope called concave function
  • a zero second derivative means a straight curve

How does Second Derivative Calculator Works?

An online double derivative calculator provides the second differentiation of given values by following these steps:

Input: 

  • First, enter an equation for the second derivative.
  • Now, select a variable for differentiation.
  • Hit the “Calculate Second Derivative” button.

Output:

  • The 2nd derivative calculator displays the step-by-step calculations for the given expression.

FAQ:

What is the second derivative test used for?

The second derivative is used to find the local extrema of the function under particular conditions. If a function f has a critical point for which f′(x) = 0 and the second derivative is positive (+ve) at this point, then the function f has a local minimum here.

When the second derivative is positive?

The sign of the second derivative tells about its concavity. If the second derivative is defined on an interval (m, n) and f ”(x) > 0 on a certain interval, then the derivative of the function is positive.

Conclusion:

Use this online second derivative calculator for second-order differentiation with complete calculations. Find the second derivative of a certain function is such a time-consuming task, but thanks to this calculator that provides all calculations quickly.

Reference: 

From the source of Wikipedia: Second order derivative, power rule, Alternative Notation, Concavity, Inflection points, Second derivative test.

From the source of Libre Text: Concavity, Tangent Lines, Test for Concavity, Point of inflection, Concave down, Concave up.

From the source of ITCC Online: Concavity and inflection Points, Definition Of Concavity, The Second Derivative Test, Inflection point.