Find the Second Derivative f(x)=12e^x-e^(2x)
The question asks for the calculation of the second derivative of the function f(x) = 12e^x - e^(2x). The second derivative refers to the derivative of the derivative, which provides information about the concavity and the acceleration of change of the original function. The problem requires applying the rules of differentiation twice to determine the rate at which the first derivative itself is changing.
$f \left(\right. x \left.\right) = 12 e^{x} - e^{2 x}$
Answer
Solution:
Step:1
Find the first derivative.
Step:1.1
Apply the Sum Rule to differentiate $f(x) = 12e^x - e^{2x}$ term by term: $\frac{d}{dx}(12e^x) + \frac{d}{dx}(-e^{2x})$.
Step:1.2
Compute $\frac{d}{dx}(12e^x)$.
Step:1.2.1
Since 12 is a constant, it can be factored out: $12\frac{d}{dx}(e^x)$.
Step:1.2.2
Use the Exponential Rule: $\frac{d}{dx}(a^x) = a^x\ln(a)$, with $a=e$, to get $12e^x$.
Step:1.3
Compute $\frac{d}{dx}(-e^{2x})$.
Step:1.3.1
Factor out the constant $-1$: $-\frac{d}{dx}(e^{2x})$.
Step:1.3.2
Apply the Chain Rule: $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$, where $f(x) = e^x$ and $g(x) = 2x$.
Step:1.3.2.1
Let $u = 2x$ and differentiate: $\frac{d}{du}(e^u)\frac{d}{dx}(2x)$.
Step:1.3.2.2
Apply the Exponential Rule: $\frac{d}{du}(e^u) = e^u$.
Step:1.3.2.3
Substitute $u$ back: $e^{2x}\frac{d}{dx}(2x)$.
Step:1.3.3
Differentiate $2x$: $2\frac{d}{dx}(x)$.
Step:1.3.4
Apply the Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, with $n=1$, to get $2$.
Step:1.3.5
Multiply $2$ by $1$: $2$.
Step:1.3.6
Combine terms: $2e^{2x}$.
Step:1.3.7
Combine the results: $f'(x) = 12e^x - 2e^{2x}$.
Step:2
Find the second derivative.
Step:2.1
Apply the Sum Rule to differentiate $f'(x) = 12e^x - 2e^{2x}$ term by term: $\frac{d}{dx}(12e^x) + \frac{d}{dx}(-2e^{2x})$.
Step:2.2
Compute $\frac{d}{dx}(12e^x)$.
Step:2.2.1
Since 12 is a constant, it can be factored out: $12\frac{d}{dx}(e^x)$.
Step:2.2.2
Use the Exponential Rule: $\frac{d}{dx}(a^x) = a^x\ln(a)$, with $a=e$, to get $12e^x$.
Step:2.3
Compute $\frac{d}{dx}(-2e^{2x})$.
Step:2.3.1
Factor out the constant $-2$: $-2\frac{d}{dx}(e^{2x})$.
Step:2.3.2
Apply the Chain Rule: $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$, where $f(x) = e^x$ and $g(x) = 2x$.
Step:2.3.2.1
Let $u = 2x$ and differentiate: $\frac{d}{du}(e^u)\frac{d}{dx}(2x)$.
Step:2.3.2.2
Apply the Exponential Rule: $\frac{d}{du}(e^u) = e^u$.
Step:2.3.2.3
Substitute $u$ back: $e^{2x}\frac{d}{dx}(2x)$.
Step:2.3.3
Differentiate $2x$: $2\frac{d}{dx}(x)$.
Step:2.3.4
Apply the Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, with $n=1$, to get $2$.
Step:2.3.5
Multiply $2$ by $1$: $2$.
Step:2.3.6
Combine terms: $4e^{2x}$.
Step:2.3.7
Combine the results: $f''(x) = 12e^x - 4e^{2x}$.
Step:3
The second derivative of $f(x)$ with respect to $x$ is $f''(x) = 12e^x - 4e^{2x}$.
Knowledge Notes:
Sum Rule: The derivative of a sum of functions is the sum of the derivatives.
Exponential Rule: The derivative of $e^x$ is $e^x$. For any base $a$, the derivative of $a^x$ is $a^x \ln(a)$.
Chain Rule: Used to differentiate composite functions. If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x))g'(x)$.
Power Rule: The derivative of $x^n$ is $nx^{n-1}$.
Constants: Constants factor out of derivatives and the derivative of a constant times a function is the constant times the derivative of the function.
Notation: $f'(x)$ denotes the first derivative of $f(x)$ with respect to $x$, and $f''(x)$ denotes the second derivative.
