Derivatives of Functions

Derivatives of Exponential and Log Functions

Differentiate f(x)=3e^x+2ln(x) using d/dx[e^x]=e^x and d/dx[ln(x)]=1/x.

exp-log-derivatives The exponential derivative is d/dx[e^x] = e^x (its own derivative). The natural log derivative is d/dx[ln(x)] = 1/x. Both combine with the sum and constant multiple rules.

Example

Apply the exact derivative rules for e to x and natural log.

highlighted = computed this step

Step 1 — Set up

Start with exponential and log terms.

f(x)=3ex+2ln⁡xf(x)= 3e^{x}+2\ln x

Step 2 — Differentiate exponential

The e to x term stays 3 e to x.

ddx3ex=3ex\frac{d}{dx} \hl{3} e^x= \hl{3} e^x

Step 3 — Differentiate logarithm

The natural-log term becomes 2 over x.

ddx2ln⁡x=2x\frac{d}{dx} \hl{2} \ln x=\frac{ \hl{2} }{x}

Step 4 — Result

Combine the exponential and log derivatives.

f′(x)=3ex+2xf'(x)= \hlmath{3e^{x}+\frac{2}{x}}