Question

Q1 : Differentiate e^2x In(x^2 - 3) with respect to x.

Q2: Differentiate (3x^2 + 2x) e^-x with respect to x.

Answer 1

Question 24847 Differentiate $e^{2x} \log(x^2 - 3)$ with respect to $x$ and differentiate $(3x^2 + 2x)e^{-x}$ with respect to $x$.

Solution. Using the rule of differentiating product $(f \cdot g)' = f' \cdot g + g' \cdot f$ and the rule of differentiating of superposition of functions $(f(g(x)))' = f'(g(x))g'(x)$ one gets that $\frac{d}{dx}(e^{2x} \log(x^2 - 3)) = 2e^{2x} \log(x^2 - 3) + e^{2x} \frac{2x}{x^2 - 3}$. Next, $((3x^2 + 2x)e^{-x})' = (6x + 2)e^{-x} - e^{-x}(3x^2 + 2x) = e^{-x}(-3x^2 + 4x + 2)$.