Exponential and logarithmic derivatives

Fact: $B\left(x\right)=b^x$ is continuous.

Fact: $B^{\prime }\left(0\right)=\underset{ℎ\to 0}{lim}\frac{b^{0+ℎ}-b^0}{ℎ}$ exists.

Fact: there is a unique value $𝒆$ such that

Examples

• $\frac{d}{dx}{ℯ}^{sin(x^2+5x)}$
• $\frac{d}{dx}\left(\frac{{ℯ}^{x^2}}{x}\right)$

• $\frac{d}{dx}ln(x^{2}sinx)$
• $\frac{d}{dx}ln\left(\frac{x^{2}sin(x^3+4x)}{2x+1}\right)$

• $\frac{d}{dx}\frac{2^x}{3^x+1}$
• $\frac{d}{dx}{log}_5(4^x+9)$

Logarithmic differentiation

• $y=x^r$
• $y=x^x$
• $y={(2x^4+1)}^{tanx}$
• $y=\frac{x^{\frac{2}{3}}\sqrt{1+x^2}}{x+x^3}$