Find the Second Derivative g(x)=8x^2e^x
The problem asks for the calculation of the second derivative of the given function g(x) = 8x^2 * e^x with respect to x. In other words, it wants you to perform differentiation twice on the function g(x) to determine the rate of change of the slope of the curve described by g(x). This process will require you to use the product rule of differentiation as well as the chain rule since the function g(x) is a product of a polynomial (8x^2) and an exponential function (e^x).
$g \left(\right. x \left.\right) = 8 x^{2} e^{x}$
Answer
Solution:
Step 1: Calculate the first derivative of $g(x)$.
Step 1.1: Apply the constant multiple rule.
The derivative of $8x^2e^x$ is $8\frac{d}{dx}(x^2e^x)$, so $g'(x) = 8\frac{d}{dx}(x^2e^x)$.
Step 1.2: Use the product rule.
The product rule gives us $g'(x) = 8(x^2\frac{d}{dx}(e^x) + e^x\frac{d}{dx}(x^2))$.
Step 1.3: Differentiate $e^x$.
The derivative of $e^x$ is $e^x$, so $g'(x) = 8(x^2e^x + e^x\frac{d}{dx}(x^2))$.
Step 1.4: Apply the power rule to $x^2$.
The power rule gives us $\frac{d}{dx}(x^2) = 2x$, so $g'(x) = 8(x^2e^x + 2xe^x)$.
Step 1.5: Simplify the expression.
Step 1.5.1: Distribute the constant $8$.
We get $g'(x) = 8x^2e^x + 8(2xe^x)$.
Step 1.5.2: Multiply $2$ by $8$.
This simplifies to $g'(x) = 8x^2e^x + 16xe^x$.
Step 1.5.3: Combine like terms.
The first derivative simplifies to $g'(x) = 8x^2e^x + 16xe^x$.
Step 1.5.4: Maintain the order of terms.
The first derivative is $g'(x) = 8x^2e^x + 16xe^x$.
Step 2: Compute the second derivative of $g(x)$.
Step 2.1: Apply the sum rule.
The derivative of $g'(x)$ is $\frac{d}{dx}(8x^2e^x) + \frac{d}{dx}(16xe^x)$.
Step 2.2: Differentiate $8x^2e^x$.
Step 2.2.1: Factor out the constant $8$.
We have $\frac{d}{dx}(8x^2e^x) = 8\frac{d}{dx}(x^2e^x)$.
Step 2.2.2: Use the product rule again.
This gives us $8(x^2\frac{d}{dx}(e^x) + e^x\frac{d}{dx}(x^2))$.
Step 2.2.3: Differentiate $e^x$.
We get $8(x^2e^x + e^x\frac{d}{dx}(x^2))$.
Step 2.2.4: Apply the power rule to $x^2$.
This results in $8(x^2e^x + 2xe^x)$.
Step 2.3: Differentiate $16xe^x$.
Step 2.3.1: Factor out the constant $16$.
We have $\frac{d}{dx}(16xe^x) = 16\frac{d}{dx}(xe^x)$.
Step 2.3.2: Apply the product rule.
This gives us $16(x\frac{d}{dx}(e^x) + e^x\frac{d}{dx}(x))$.
Step 2.3.3: Differentiate $e^x$.
We get $16(xe^x + e^x\frac{d}{dx}(x))$.
Step 2.3.4: Apply the power rule to $x$.
Since $\frac{d}{dx}(x) = 1$, we have $16(xe^x + e^x)$.
Step 2.3.5: Simplify the expression.
The result is $16xe^x + 16e^x$.
Step 2.4: Combine and simplify all terms.
Step 2.4.1: Distribute the constants.
We get $8x^2e^x + 16xe^x + 16xe^x + 16e^x$.
Step 2.4.2: Combine like terms.
The second derivative is $8x^2e^x + 32xe^x + 16e^x$.
Step 2.4.3: Maintain the order of terms.
The final form of the second derivative is $g''(x) = 8x^2e^x + 32xe^x + 16e^x$.
Step 3: Conclude the second derivative.
The second derivative of $g(x)$ is $g''(x) = 8x^2e^x + 32xe^x + 16e^x$.
Knowledge Notes:
Constant Multiple Rule: If $c$ is a constant and $f(x)$ is a differentiable function, then the derivative of $cf(x)$ is $c\frac{d}{dx}f(x)$.
Product Rule: For differentiable functions $f(x)$ and $g(x)$, the derivative of their product $f(x)g(x)$ is given by $f'(x)g(x) + f(x)g'(x)$.
Power Rule: If $n$ is a real number, then the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Exponential Rule: The derivative of $e^x$ with respect to $x$ is $e^x$.
Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
Simplification: Combining like terms and simplifying expressions are standard algebraic techniques used to present the derivative in its simplest form.
