Calculus Examples

Find the Second Derivative 5x^(-1/2)
Step 1
Find the first derivative.
Tap for more steps...
Step 1.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2
Differentiate using the Power Rule which states that is where .
Step 1.3
To write as a fraction with a common denominator, multiply by .
Step 1.4
Combine and .
Step 1.5
Combine the numerators over the common denominator.
Step 1.6
Simplify the numerator.
Tap for more steps...
Step 1.6.1
Multiply by .
Step 1.6.2
Subtract from .
Step 1.7
Move the negative in front of the fraction.
Step 1.8
Combine and .
Step 1.9
Multiply by .
Step 1.10
Combine and .
Step 1.11
Simplify the expression.
Tap for more steps...
Step 1.11.1
Move to the denominator using the negative exponent rule .
Step 1.11.2
Move the negative in front of the fraction.
Step 2
Find the second derivative.
Tap for more steps...
Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Apply basic rules of exponents.
Tap for more steps...
Step 2.2.1
Rewrite as .
Step 2.2.2
Multiply the exponents in .
Tap for more steps...
Step 2.2.2.1
Apply the power rule and multiply exponents, .
Step 2.2.2.2
Multiply .
Tap for more steps...
Step 2.2.2.2.1
Combine and .
Step 2.2.2.2.2
Multiply by .
Step 2.2.2.3
Move the negative in front of the fraction.
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
To write as a fraction with a common denominator, multiply by .
Step 2.5
Combine and .
Step 2.6
Combine the numerators over the common denominator.
Step 2.7
Simplify the numerator.
Tap for more steps...
Step 2.7.1
Multiply by .
Step 2.7.2
Subtract from .
Step 2.8
Move the negative in front of the fraction.
Step 2.9
Combine and .
Step 2.10
Multiply.
Tap for more steps...
Step 2.10.1
Multiply by .
Step 2.10.2
Multiply by .
Step 2.11
Multiply by .
Step 2.12
Multiply.
Tap for more steps...
Step 2.12.1
Multiply by .
Step 2.12.2
Multiply by .
Step 2.12.3
Move to the denominator using the negative exponent rule .