Q.Differentiate $y = 2e^{5x} - 3\ln x$ with respect to $x$.

The exponential term needs the chain rule layered on the standard $e^x$ derivative (since the exponent is $5x$, not $x$ alone); the log term uses the standard $\ln x$ derivative directly.

Given: $y = 2e^{5x}-3\ln x$.

Step 1 — Differentiate $2e^{5x}$: using $\dfrac{d}{dx}(e^{g(x)})=e^{g(x)}g'(x)$ with $g(x)=5x$, $g'(x)=5$: $\dfrac{d}{dx}(e^{5x})=5e^{5x}$, so $\dfrac{d}{dx}(2e^{5x})=10e^{5x}$.

Step 2 — Differentiate $-3\ln x$: using $\dfrac{d}{dx}(\ln x)=\dfrac1x$: $\dfrac{d}{dx}(-3\ln x) = -\dfrac3x$.

Step 3 — Combine: $\dfrac{dy}{dx} = 10e^{5x}-\dfrac3x$. …

Differentiating $e^{5x}$ as simply $e^{5x}$ (forgetting to multiply by the derivative of the exponent, $5$, via the chain rule) — the standard-function result $\frac{d}{dx}(e^x …