Find dy/dx y=e^(-7x)sin(x)
The problem is asking for the derivative of the function \( y = e^{-7x}\sin(x) \) with respect to \( x \). In other words, you need to calculate the rate at which the function changes as \( x \) changes, which involves applying calculus concepts such as the chain rule, the product rule, or both, to differentiate the given function correctly. The solution to this problem should provide the expression for \( \frac{dy}{dx} \), which represents the slope of the tangent to the curve described by the function at any given point \( x \).
$y = e^{- 7 x} sin \left(\right. x \left.\right)$
Answer
Solution:
Step:1
Apply differentiation to both sides of the equation $y = e^{-7x}\sin(x)$: $\frac{d}{dx}(y) = \frac{d}{dx}(e^{-7x}\sin(x))$.
Step:2
The derivative of $y$ with respect to $x$ is denoted by $\frac{dy}{dx}$.
Step:3
Proceed to differentiate the expression on the right-hand side.
Step:3.1
Invoke the Product Rule for differentiation: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = e^{-7x}$ and $v = \sin(x)$. Thus, we have $e^{-7x}\frac{d}{dx}(\sin(x)) + \sin(x)\frac{d}{dx}(e^{-7x})$.
Step:3.2
Compute the derivative of $\sin(x)$, which is $\cos(x)$: $e^{-7x}\cos(x) + \sin(x)\frac{d}{dx}(e^{-7x})$.
Step:3.3
Apply the Chain Rule for differentiation, which states that $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$. Here, $f(x) = e^x$ and $g(x) = -7x$.
Step:3.3.1
Set $u = -7x$ to apply the Chain Rule: $e^{-7x}\cos(x) + \sin(x)\left(\frac{d}{du}(e^u)\frac{d}{dx}(-7x)\right)$.
Step:3.3.2
Use the Exponential Rule of differentiation: $\frac{d}{du}(e^u) = e^u\ln(e)$, where $a = e$: $e^{-7x}\cos(x) + \sin(x)(e^u\frac{d}{dx}(-7x))$.
Step:3.3.3
Replace $u$ with $-7x$: $e^{-7x}\cos(x) + \sin(x)(e^{-7x}\frac{d}{dx}(-7x))$.
Step:3.4
Perform the differentiation.
Step:3.4.1
Since $-7$ is a constant, its derivative with respect to $x$ is $-7$: $e^{-7x}\cos(x) + \sin(x)(e^{-7x}(-7))$.
Step:3.4.2
Apply the Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 1$: $e^{-7x}\cos(x) + \sin(x)(e^{-7x}(-7 \cdot 1))$.
Step:3.4.3
Simplify the expression.
Step:3.4.3.1
Multiply $-7$ by $1$: $e^{-7x}\cos(x) + \sin(x)(-7 \cdot e^{-7x})$.
Step:3.4.3.2
Rearrange terms to group like factors: $e^{-7x}\cos(x) - 7e^{-7x}\sin(x)$.
Step:4
Combine the terms to form the final expression for the derivative: $y' = e^{-7x}\cos(x) - 7e^{-7x}\sin(x)$.
Step:5
Substitute $\frac{dy}{dx}$ for $y'$ to complete the differentiation: $\frac{dy}{dx} = e^{-7x}\cos(x) - 7e^{-7x}\sin(x)$.
Knowledge Notes:
Product Rule: The Product Rule is a formula used to find the derivative of the product of two functions. It is given by $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$.
Chain Rule: The Chain Rule is used to differentiate composite functions. If $y = f(g(x))$, then the derivative of $y$ with respect to $x$ is $\frac{dy}{dx} = f'(g(x))g'(x)$.
Exponential Rule: This rule states that the derivative of $e^u$ with respect to $u$ is $e^u\ln(e)$, which simplifies to $e^u$ since $\ln(e) = 1$.
Power Rule: The Power Rule is used to differentiate functions of the form $x^n$, where $n$ is any real number. The rule states that $\frac{d}{dx}(x^n) = nx^{n-1}$.
Constants in Differentiation: When differentiating, any constant multiplier can be taken outside the differentiation operator. For example, $\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))$, where $c$ is a constant.
Simplification: After applying the differentiation rules, it is often necessary to simplify the expression to combine like terms and present the derivative in its simplest form.
