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Question

# Use the first and second derivatives to explain the properties of the graph of $\cosh x$.

Solution

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As always, the first derivatives provides us with intervals where $f$ increases and decreases, while the second derivative tells us where $f$ is concave up or down and where to find points of inflexion.

We already know the first derivative:

$f(x)=\cosh x\Rightarrow f^\prime(x)=\sinh x\text{.}$

Now we use the chain rule to find the second derivative

$f^{\prime\prime}(x)=\left( \sinh x\right)^\prime=\cosh x\text{.}$

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