Determine higher-order derivatives

Repeated differentiation produces the second, third, and, when a clear pattern exists, higher-order derivatives, written $f''(x)$ , $f'''(x)$ , and $f^{(n)}(x)$ , using familiar differentiation rules for polynomial, exponential, logarithmic, and trigonometric functions. The learner interprets $f''$ as the rate of change of the first derivative, including concavity and acceleration when $f$ represents position. This scope excludes abstract higher-order formulas, multivariable derivatives, and advanced generalized differentiation rules.

Detailed Explanation: Determine higher-order derivatives

To find a higher-order derivative, differentiate repeatedly. Each new derivative is found by differentiating the derivative before it:

$f'(x)$ : first derivative
$f''(x)$ : second derivative
$f'''(x)$ : third derivative
$f^{(4)}(x)$ : fourth derivative

Use the usual differentiation rules each time.

Worked example

Find the first five derivatives of

f(x)=2x^4-3x^3+5x^2-7x+1.

Step 1: Find the first derivative

Differentiate each term:

f'(x)=8x^3-9x^2+10x-7.

Step 2: Find the second derivative

Differentiate $f'(x)$ :

f''(x)=24x^2-18x+10.

The second derivative tells us how quickly the first derivative is changing.

Step 3: Find the third derivative

Differentiate $f''(x)$ :

f'''(x)=48x-18.

Step 4: Find the fourth derivative

Differentiate again:

f^{(4)}(x)=48.

Step 5: Find the fifth derivative

The derivative of a constant is zero:

f^{(5)}(x)=0.

Therefore, the derivatives are

\boxed{f'(x)=8x^3-9x^2+10x-7}

\boxed{f''(x)=24x^2-18x+10}

\boxed{f'''(x)=48x-18}

\boxed{f^{(4)}(x)=48}

\boxed{f^{(5)}(x)=0}

When $f(x)$ represents position, $f'(x)$ is velocity and $f''(x)$ is acceleration. Also, if $f''(x)>0$ , the graph of $f$ is concave up; if $f''(x)<0$ , it is concave down.

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Critical Points - Function and Interval to Concavity

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