Question

find the derivative of the function (y = e^{4x^{2}+3x - 4}) using the chain rule for exponential functions. (\frac{dy}{dx}=)

Explanation:

Step1: Identify outer and inner functions

Let $u = 4x^{2}+3x - 4$, so $y = e^{u}$.

Step2: Differentiate outer function

The derivative of $y = e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$.

Step3: Differentiate inner function

The derivative of $u = 4x^{2}+3x - 4$ with respect to $x$ is $\frac{du}{dx}=(4\times2x)+3=8x + 3$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$. Substitute $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=8x + 3$ back in. Since $u = 4x^{2}+3x - 4$, we get $\frac{dy}{dx}=e^{4x^{2}+3x - 4}(8x + 3)$.

Answer:

$(8x + 3)e^{4x^{2}+3x - 4}$