Differentiating Exponentials and Logarithms
Rules
- If \(y=e^{kx}\), then \(\frac{dy}{dx}=ke^{kx}\)
- If \(y=\ln x\) then \(\frac{dy}{dx}=\frac{1}{x}\)
- If \(y=a^x\), then \(\frac{dy}{dx}=a^x\ln a\)
Differentiation
- a) \(e^x=e^x\)
b) \(3e^x=3e^x\)
c) \(\ln x=\frac{1}{x}\)
d) \(\frac{1}{2}\ln x=\frac{1}{2x}\)
2.
a) \(7-2e^t=-2e^t\)
b) \(3t^2+\ln t=6t+\frac{1}{t}\)
c) \(e^t+t^5=e^t+5t^4\)
Book Questions
- a) \(y=4e^{7x}\to28e^{7x}\)
b) \(y=3^x\to{3}^x\ln3\)
c) \(y=\left( \frac{1}{2} \right)^x\to\left( \frac{1}{2} \right)^x \ln\frac{1}{2}\)
d) \(y=\ln5x\to\frac{1}{x}\)
e) \(y=4\left( \frac{1}{3} \right)^x\to4\left( \frac{1}{3} \right)^x\ln\frac{1}{3}\)
f)