Question

find f(x). f(x)=3 ln x + 4e^x - 5x^2 f(x)=□

Explanation:

Step1: Differentiate each term separately

We know that the derivative of $\ln x$ is $\frac{1}{x}$, the derivative of $e^x$ is $e^x$ and the derivative of $x^n$ is $nx^{n - 1}$.

Step2: Differentiate $3\ln x$

Using the constant - multiple rule and the derivative of $\ln x$, we have $\frac{d}{dx}(3\ln x)=3\times\frac{1}{x}=\frac{3}{x}$.

Step3: Differentiate $4e^x$

Using the constant - multiple rule and the derivative of $e^x$, we get $\frac{d}{dx}(4e^x)=4e^x$.

Step4: Differentiate $- 5x^2$

Using the constant - multiple rule and the power rule, we have $\frac{d}{dx}(-5x^2)=-5\times2x=-10x$.

Step5: Combine the derivatives

$f'(x)=\frac{d}{dx}(3\ln x)+\frac{d}{dx}(4e^x)+\frac{d}{dx}(-5x^2)=\frac{3}{x}+4e^x - 10x$.

Answer:

$\frac{3}{x}+4e^x - 10x$