First Derivatives of eˣ and ln x | CAIE 9709 Pure Mathematics 3 Notes

Pure Mathematics 3

Mathematics9709

The natural exponential

Definition

What is meant by natural exponential?

Model answer: $e^x$ is the exponential function with base $e\approx2.718$ . It is the special exponential whose gradient at every $x$ equals its height.

\frac{d}{dx}(e^x)=e^x

The value of the exponential is always positive: $e^x>0$ for every real $x$ . It can never be a zero factor in a stationary equation.

The natural logarithm

Definition

What is meant by natural logarithm?

Model answer: $y=\ln x$ means $e^y=x$ . Therefore its real domain is $x>0$ .

The natural logarithm is the inverse of the natural exponential: $\ln(e^y)=y$ and $e^{\ln x}=x$ for $x>0$ .

\frac{d}{dx}(\ln x)=\frac1x\qquad(x>0)

Common mistake

\ln x

is not a power of

x

. Do not use the power rule on it.

Your turnIndependent check [4 marks]

Differentiate: (a) $7e^x-3x^2$ ; (b) $4\ln x+x^{-1}$ . State the domain needed for part (b).

Show worked answer

Answer

\text{(a)}\ 7e^x-6x

\text{(b)}\ \frac4x-\frac1{x^2},\quad x>0

The natural exponential

Definition

What is meant by natural exponential?

Model answer: $e^x$ is the exponential function with base $e\approx2.718$ . It is the special exponential whose gradient at every $x$ equals its height.

\frac{d}{dx}(e^x)=e^x

The value of the exponential is always positive: $e^x>0$ for every real $x$ . It can never be a zero factor in a stationary equation.

The natural logarithm

Definition

What is meant by natural logarithm?

Model answer: $y=\ln x$ means $e^y=x$ . Therefore its real domain is $x>0$ .

The natural logarithm is the inverse of the natural exponential: $\ln(e^y)=y$ and $e^{\ln x}=x$ for $x>0$ .

\frac{d}{dx}(\ln x)=\frac1x\qquad(x>0)

Common mistake

\ln x

is not a power of

x

. Do not use the power rule on it.

Your turnIndependent check [4 marks]

Differentiate: (a) $7e^x-3x^2$ ; (b) $4\ln x+x^{-1}$ . State the domain needed for part (b).

Show worked answer

Answer

\text{(a)}\ 7e^x-6x

\text{(b)}\ \frac4x-\frac1{x^2},\quad x>0