Unit 6

Exponential and Logarithmic Functions

Lesson 1: Derivatives of Exponential Functions

Imagine a function such that $f\mathit{\text{'}}\mathrm{(}x\mathrm{)}=f\left(x\right)$. That would mean $f\mathit{\text{'}}\left(3\right)=f\left(3\right)$ and $f\mathit{\text{'}}\left(-5\right)=f\left(-5\right)$. What might the graph of such a function look like?

If visualizing this type of graph is a challenge, try breaking it down as follows.

For a given $x$-value, $f\left(x\right)$ represents the corresponding $y$-value on the graph and $f\mathit{\text{'}}\mathrm{(}x\mathrm{)}$ represents the slope of the line tangent to the curve through the point $\left(x\mathrm{,} y\right)$. So, if $f\mathit{\text{'}}\mathrm{(}x\mathrm{)}=f\left(x\right)$, the $y$-value (the function value) and the slope of the function will be equal. Take a moment to try to sketch the graph of a function whose $y$-values and slopes are the same for any given $x$ on the domains of the function and its derivative function.

• Can you sketch the graph of a function such that $f\mathit{\text{'}}\mathrm{(}x\mathrm{)}=f\left(x\right)$?
• What kind of function could satisfy the condition $f\mathit{\text{'}}\mathrm{(}x\mathrm{)}=f\left(x\right)$?

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