Calculus Examples

Find the Second Derivative f(x)=e^(1/x)
Step 1
Find the first derivative.
Tap for more steps...
Step 1.1
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 1.1.1
To apply the Chain Rule, set as .
Step 1.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.3
Replace all occurrences of with .
Step 1.2
Differentiate using the Power Rule.
Tap for more steps...
Step 1.2.1
Rewrite as .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.3
Simplify.
Tap for more steps...
Step 1.3.1
Rewrite the expression using the negative exponent rule .
Step 1.3.2
Combine and .
Step 2
Find the second derivative.
Tap for more steps...
Step 2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2
Differentiate using the Quotient Rule which states that is where and .
Step 2.3
Multiply the exponents in .
Tap for more steps...
Step 2.3.1
Apply the power rule and multiply exponents, .
Step 2.3.2
Multiply by .
Step 2.4
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 2.4.1
To apply the Chain Rule, set as .
Step 2.4.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.4.3
Replace all occurrences of with .
Step 2.5
Differentiate using the Power Rule.
Tap for more steps...
Step 2.5.1
Rewrite as .
Step 2.5.2
Differentiate using the Power Rule which states that is where .
Step 2.6
Multiply by by adding the exponents.
Tap for more steps...
Step 2.6.1
Move .
Step 2.6.2
Use the power rule to combine exponents.
Step 2.6.3
Add and .
Step 2.7
Simplify the expression.
Tap for more steps...
Step 2.7.1
Simplify .
Step 2.7.2
Move to the left of .
Step 2.7.3
Rewrite as .
Step 2.8
Differentiate using the Power Rule which states that is where .
Step 2.9
Multiply by .
Step 2.10
Since is constant with respect to , the derivative of with respect to is .
Step 2.11
Simplify the expression.
Tap for more steps...
Step 2.11.1
Multiply by .
Step 2.11.2
Add and .
Step 2.12
Simplify.
Tap for more steps...
Step 2.12.1
Reorder factors in .
Step 2.12.2
Factor out of .
Tap for more steps...
Step 2.12.2.1
Factor out of .
Step 2.12.2.2
Factor out of .
Step 2.12.2.3
Factor out of .
Step 2.12.3
Rewrite as .
Step 2.12.4
Factor out of .
Step 2.12.5
Factor out of .
Step 2.12.6
Move the negative in front of the fraction.
Step 2.12.7
Multiply by .
Step 2.12.8
Multiply by .
Step 3
The second derivative of with respect to is .