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Wize University Calculus 1 Textbook > Derivatives

Higher Order Derivatives

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Higher Order Derivatives

To get the second derivative, differentiate the first derivative. Then the third is the derivative of the second and so on.

The second derivative is denoted:

f′′(x)=y′′=d2fdx2=d2ydx2=d2dx2f(x)=ddx(dfdx)\boxed{\displaystyle f''(x) =y'' =\frac{d^2f}{dx^2} =\frac{d^2y}{dx^2} =\frac{d^2}{dx^2}f\left(x\right)=\frac{d}{dx}\left(\frac{df}{dx}\right)}

The nth derivative is denoted:

fn(x)=dnfdxn \boxed{\displaystyle f^n(x) =\frac{d^nf}{dx^n}\ }



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Example: Higher Order Derivatives

Find f′(π)+f′′(π)f'\left(\pi\right)+f''\left(\pi\right) if f(x)=x cos⁡x−sin⁡xf\left(x\right)=x\ \cos x-\sin x.

First Derivative
There's not much we can simplify with this expression, and since we have a product of functions, we need to use product rule.
f′(x)=(1)(cos⁡x)+(−sin⁡x)(x)−cos⁡xf'\left(x\right)=\left(1\right)\left(\cos x\right)+\left(-\sin x\right)\left(x\right)-\cos x
f′(x)=cos⁡x−x sin⁡x−cos⁡xf'\left(x\right)=\cos x-x\ \sin x-\cos x
f′(x)=−x sin⁡xf'\left(x\right)=-x\ \sin x

Second Derivative
Since we have a product of functions, we need the product rule again.
f′′(x)=(−1)(sin⁡x)+(cos⁡x)(−x)f''\left(x\right)=\left(-1\right)\left(\sin x\right)+\left(\cos x\right)\left(-x\right)
f′′(x)=−sin⁡x−xcos⁡xf''\left(x\right)=-\sin x-x\cos x

Substituting x=πx= \pi:
f′(π)+f′′(π)f'\left(\pi\right)+f''\left(\pi\right)
=−πsin⁡π+(−sin⁡π−πcos⁡π)=-\pi\sin\pi+\left(-\sin\pi-\pi\cos\pi\right)
=0+(0−π(−1))=0+\left(0-\pi\left(-1\right)\right)
=π=\pi
Find the third derivative of f(x)=x2+16xf(x) = x^2+16\sqrt{x}.
If f(x)=x10f\left(x\right)=x^{10} find f(9)(x)f^{\left(9\right)}\left(x\right).
(i.e. the 9th derivative)
Extra Practice
Higher Order Derivatives: The Power Rule
If f(x)=x10f\left(x\right)=x^{10} find f(9)(x)f^{\left(9\right)}\left(x\right).
(i.e. the 9th derivative)
Higher Order Derivatives
Find the second derivative of f(x)=x2+16xf(x) = x^2+16\sqrt{x}.
Higher order derivatives
Consider the function f(x)=3x−4xf(x)=\dfrac{3x-4}{x}. Find the second derivative f′′(x), at x=1f''(x),\text{ at }x=1.
Higher order derivatives
Consider the function f(x)=3x−4xf(x)=\dfrac{3x-4}{x}. Find the second derivative f′′(x), at x=1f''(x),\text{ at }x=1.
The Chain Rule
If 2−g(x)=x2+2[f(x)]2−x3g(x)2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right), f(0)=1 , f′(0)=−2, g(0)=−1 and  g′′(0)=3f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3, then the value of f′′(0)f''\left(0\right) is equal to
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Higher order derivatives
Consider the function f(x)=3x−4xf(x)=\dfrac{3x-4}{x}. Find the second derivative f′′(x), at x=1f''(x),\text{ at }x=1.
Higher Order Derivatives
An object is thrown straight up in the air at t=0t = 0 seconds. Its height in metres at tt seconds is given by
h(t)=s0+v0t−7t2h(t) = s_0 + v_0 t - 7t^2
where s0s_0 and v0v_0 are constants. In the first second the object rises 7 metres. For how many seconds does the object rise before starting to fall back down?
The Chain Rule
If 2−g(x)=x2+2[f(x)]2−x3g(x)2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right), f(0)=1 , f′(0)=−2, g(0)=−1 and  g′′(0)=3f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3, then the value of f′′(0)f''\left(0\right) is equal to
Higher Order Derivatives: The Power Rule
If f(x)=x10f\left(x\right)=x^{10} find f(9)(x)f^{\left(9\right)}\left(x\right).
(i.e. the 9th derivative)
Higher Order Derivatives
Find the third derivative of f(x)=x2+16xf(x) = x^2+16\sqrt{x}.
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Higher order derivatives
Consider the function f(x)=3x−4xf(x)=\dfrac{3x-4}{x}. Find the second derivative f′′(x), at x=1f''(x),\text{ at }x=1.