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Wize High School Grade 12 Calculus Textbook > Derivatives of Exponential (& Logarithmic) Functions

Review of Exponential Functions y=exy=e^x

Derivative of Exponential Functions y=bxy=b^xPrevious Section
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Review of Exponential Functions y = ex

An exponential function has the form f(x)=bxf\left(x\right)=b^x for any positive real number bb, b≠1b\ne1.

Recall: The natural number e≈2.718...e\approx2.718...

When the base b=eb=e we get a special exponential function f(x)=exf\left(x\right)=e^x.

Properties
  • f(x)f\left(x\right) has a horizontal asymptote at y=0y=0
  • The y-intercept is (0, 1)\left(0,\ 1\right)
  • Domain: x∈Rx\in R; Range: y>0y>0

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Inverse Function -- f(x)=log⁡ex=ln⁡xf\left(x\right)=\log_ex=\ln x
  • y=ln⁡x   ↔   x=ey\boxed{y=\ln x\ \ \ \leftrightarrow\ \ \ x=e^y}
  • eln⁡x=x\boxed{e^{\ln x}=x} and ln⁡(ex)=x\boxed{\ln\left(e^x\right)=x}
  • The x-intercept is (1, 0)\left(1,\ 0\right)
  • Domain: x>0x>0; Range: y∈Ry\in R
  • It has a vertical asymptote at x=0x=0

Note:
lim⁡h→0 bh−1h=ln⁡b\displaystyle\lim_{h\to0}\ \frac{b^h-1}{h}=\ln b

Practice: e & ex

The number ee is defined by
  • e=lim⁡n→∞(1+1n)n\displaystyle e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n and
  • ex=lim⁡n→∞(1+xn)n\displaystyle e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n
Using these definitions, estimate the following.