Higher-Order Derivative Calculator

Calculate the nth derivative of a function step by step, including second, third, fourth, and higher derivatives.

Enter a function \(f(x)\), choose the derivative order, and calculate the result symbolically.

What Is a Higher-Order Derivative?

A higher-order derivative is obtained by differentiating a function more than once. The first derivative is written as \(f'(x)\), the second derivative as \(f''(x)\), and the \(n\)-th derivative as \(f^{(n)}(x)\).

For example, consider:

\[ f(x)=x^4 \] \[ f'(x)=4x^3 \] \[ f''(x)=12x^2 \] \[ f'''(x)=24x \] \[ f^{(4)}(x)=24 \] \[ f^{(5)}(x)=0 \]

A polynomial eventually becomes zero after enough derivatives. Other functions, such as \(e^x\), can continue producing non-zero derivatives indefinitely.

Second Derivative

The second derivative is the derivative of the first derivative:

\[ f''(x) = \frac{d}{dx}\left(f'(x)\right) = \frac{d^2f}{dx^2} \]

The second derivative is frequently used to study the concavity of a function. In applications, it can also describe quantities such as acceleration when the first derivative represents velocity.

Third and Higher Derivatives

The third derivative is obtained by differentiating the second derivative:

\[ f'''(x) = \frac{d}{dx}\left(f''(x)\right) \]

More generally, the \(n\)-th derivative is:

\[ f^{(n)}(x) = \frac{d^n f}{dx^n} \]

Higher-order derivatives appear in Taylor series, differential equations, physics, engineering, mathematical modeling, and higher-order approximations.

Higher-Order Power Rule

For a power function \(x^m\), the first derivative is:

\[ \frac{d}{dx}x^m = mx^{m-1} \]

Repeated application gives:

\[ \frac{d^n}{dx^n}x^m = \frac{m!}{(m-n)!}x^{m-n} \]

For a polynomial with a non-negative integer exponent, the derivative eventually becomes zero once the derivative order exceeds the polynomial's degree.

Common Higher-Order Derivative Patterns

Exponential Function

The exponential function \(e^x\) is unchanged when differentiated:

\[ \frac{d^n}{dx^n}e^x=e^x \]

Sine

The derivatives of sine repeat in a four-step cycle:

\[ \sin(x) \rightarrow \cos(x) \rightarrow -\sin(x) \rightarrow -\cos(x) \rightarrow \sin(x) \]

Cosine

Cosine follows a similar four-step cycle:

\[ \cos(x) \rightarrow -\sin(x) \rightarrow -\cos(x) \rightarrow \sin(x) \rightarrow \cos(x) \]

Higher-Order Derivative Examples

Example 1: Polynomial

For \(f(x)=x^3\):

\[ f'(x)=3x^2 \] \[ f''(x)=6x \] \[ f'''(x)=6 \] \[ f^{(4)}(x)=0 \]

Example 2: Sine

For \(f(x)=\sin(x)\):

\[ f'(x)=\cos(x) \] \[ f''(x)=-\sin(x) \] \[ f'''(x)=-\cos(x) \] \[ f^{(4)}(x)=\sin(x) \]

Supported Input

The calculator currently supports the following symbolic forms:

Products of functions, nested functions, and general chain-rule expressions are not currently supported. Use the formats listed above for reliable results.

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Frequently Asked Questions

What is a higher-order derivative?

A higher-order derivative is a derivative taken more than once. For example, \(f''(x)\) is the second derivative, \(f'''(x)\) is the third derivative, and \(f^{(n)}(x)\) is the nth derivative.

What is an nth derivative?

The nth derivative is the result of differentiating a function \(n\) times. For example, selecting order 4 calculates \(f^{(4)}(x)\).

How do I calculate a second derivative?

Enter your function and set the derivative order to 2. The calculator will show both the first and second derivatives.

How do I calculate a third derivative?

Enter your function and set the derivative order to 3. The calculator will calculate the first, second, and third derivatives in sequence.

What happens when a polynomial is differentiated many times?

A polynomial eventually becomes zero. For example, \(x^3\) becomes \(6\) after three derivatives and \(0\) after the fourth derivative.

Can higher-order derivatives be used in physics?

Yes. Higher-order derivatives occur in many areas of physics and engineering. For example, if position is differentiated once with respect to time it gives velocity, and differentiating again gives acceleration.