📘 differential calculus

1. Problem: $f(x)=\ln(2x^3)$. 2. Use the chain rule: if $f(x)=\ln(u)$ then $f'(x)=\frac{u'}{u}$.

1. A a) Given $f(x)=\ln(2x^3)$. Use the chain rule with $u=2x^3$ so $u'=6x^2$.

1. **Evaluate** \(\lim_{x \to 0} \frac{a^{x} - b^{x}}{x}\). Step 1: Recall the exponential limit property: \(\lim_{x \to 0} \frac{c^{x} - 1}{x} = \ln c\).

1. Find $\frac{d}{dx} \left(2x^2 - 5x\right)^3$. Apply the chain rule: let $u = 2x^2 - 5x$, then $f(x) = u^3$.