Subject: Calculus

Higher Order Derivatives


So far, we have encountered several types of derivatives and techniques but all those things fall under first order derivative. In this section, we will learn more about higher order derivatives. Higher order derivative is basically differentiating a differentiable function f more than once. For example, a function f has f' as its first derivative. If f' is differentiable, then its derivative is called the second order derivative of f. Again, if the second order derivative of f is differentiable, then its derivative is called third order derivative and so on. So as long as the derivative of a derivative is differentiable, then the trend must go on. Commonly, the second order derivative of function f is denoted by f'', the third order derivative is denoted by f''' and so on.
There are many notations of higher order derivatives. The table below summarizes some commonly used notations for higher order derivatives supposing that the function to be differentiated is f(x)=y.

Higher Order Derivative Notations
First OrderSecond OrderThird Ordern^{th} order
f'f''f'''f^{n}
\frac{dy}{dx}\frac{d^{2}y}{dx^{2}}\frac{d^{3}y}{dx^{3}}\frac{d^{n}y}{dx^{n}}
y'y''y'''y^{n}
D_{x}D_{x}^{2}D_{x}^{3}D_{x}^{n}

We now know the different notations involving higher order derivatives. The question is, how is it done? Don't worry, it would not be difficult. We only have to differentiate a given function n times and the result is now the n^{th} order derivative. So if we are asked to find the second order derivative of a function, we simply just differentiate that function twice and for its third order derivative, we will also just take its third order derivative and so on.

Illustration

For example, we have a function f(x)=2x^{3}. Lets try to differentiate f(x) four times. Thus,

First Derivative

f'(x)=D_{x}2x^{3}=3\cdot 2x^{2}\cdot (1)=6x^{2}

Second Derivative

f''(x)=D_{x}6x^{2}=2\cdot 6x^{1}\cdot (1)=12x

Third Derivative

f'''(x)=D_{x}12x=12

Fourth Derivative

f''''(x)=D_{x}12=0

By iterating the process of differentiation to the function f four times, we now have the value of the fourth order derivative of f which is 0 while its second and third order derivative is 12x and 12 respectively.

Example #1

Find the 3^{rd} order derivative of the function f(x)=3x^{3}

Solution: First, we differentiate once. Thus,

f'(x)=D_{x}3x^{3}
 =3\cdot 3x^{2}\cdot (1)$}
 =9x^{2}.

Then, we differentiate f'(x) to get the second order derivative which is given by

f(x)=D_{x}[f'(x)]
 =D_{x}9x^{2}
 =2\cdot 9x\cdot (1)
 =18x.

Lastly, we differentiate the f'' to get the third order derivative of f. Thus,

f'''(x)=D_{x}[f''(x)]
 =D_{x}18x
 =18.

Therefore, the third order derivative of the function f(x)=3x^{3} is 18. Let us find another example.

Example #2

Solve the following higher order derivative: Find \frac{d^{2}y}{dx^{2}} if y=\cos{2x^{2}}.

Solution: Doing the same procedure as in the above example, we find that

y'=D_{x}\cos{2x^{2}}
 =-\sin{2x^{2}}\cdot D_{x}2x^{2}
 =-4x\sin{2x^{2}}.

After that, using the product rule discussed in the previous section, we have the second derivative of y=\cos{2x^{2}} as,

y''=D_{x}[y']
 =D_{x}(-4x\sin{2x^{2}})
 =-4(x\cdot D_{x}\sin{2x^{2}}+\sin{2x^{2}}\cdot D_{x}(x)
 =-4(x\cdot \cos(2x^{2})\cdot D_{x}2x^{2}+ \sin{2x^{2}}\cdot (1))
 =-4(4x^{2}\cdot \cos{2x^{2}}+\sin{2x^{2}}).

Thus, the second order derivative of the function y=\cos{2x^{2}} is y= -4(4x^{2}\cdot \cos{2x^{2}}+\sin{2x^{2}}) .

Example #3

What is the fifth-order derivative of the function f(x)=x^5.

Answer: 120

Example #4

Solve for the n^{th}-order derivative of f(z)=e^{z}.

Answer: e^z

Example #5

Solve the fourth-order derivative of f(x)=\cos{x}.

Answer: \cos{x}

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