We are given that (dy)/(dx)=e^(5y). Find an expression for (d^(2)y)/(dx^(2)) in terms of x and y. (d^(2)y)/(dx^(2))=

Given derivative: We are given the first derivative of y with respect to x as (dy)/(dx) = e^(5y). To find the second derivative (d^(2)y)/(dx^(2)), we need to differentiate (dy)/(dx) with respect to x.Apply chain rule: Using the chain rule, we differentiate e^(5y) with respect to x. Since e^(5y) is a function of y, and y is a function of x, we apply the chain rule as follows: (d/dx)(e^(5y)) = (d/dy)(e^(5y)) * (dy/dx).Differentiate with respect to y: Differentiate e^(5y) with respect to y to get 5e^(5y). So, (d/dx)(e^(5y)) = 5e^(5y) * (dy/dx).Substitute given derivative: Substitute the given (dy)/(dx) = e^(5y) into the expression. We get (d/dx)(e^(5y)) = 5e^(5y) * e^(5y).Simplify the expression: Simplify the expression by combining the exponential terms. We have (d^(2)y)/(dx^(2)) = 5e^(5y) * e^(5y) = 5e^(10y).

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We are given that
(dy)/(dx)=e^(5y).
Find an expression for
(d^(2)y)/(dx^(2)) in terms of
x and
y.

(d^(2)y)/(dx^(2))=

We are given that $\frac{d y}{d x}=e^{5 y}$ . $\newline$ Find an expression for $\frac{d^{2} y}{d x^{2}}$ in terms of $x$ and $y$ . $\newline$ $\frac{d^{2} y}{d x^{2}}=$

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Algebra 2

Find the vertex of the transformed function

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Q. We are given that

\frac{d y}{d x}=e^{5 y}

\newline

Find an expression for

\frac{d^{2} y}{d x^{2}}

in terms of

x

and

y

\newline

\frac{d^{2} y}{d x^{2}}=

Given derivative: We are given the first derivative of $y$ with respect to $x$ as $\frac{dy}{dx} = e^{5y}$ . To find the second derivative $\frac{d^{2}y}{dx^{2}}$ , we need to differentiate $\frac{dy}{dx}$ with respect to $x$ .
Apply chain rule: Using the chain rule, we differentiate $e^{5y}$ with respect to $x$ . Since $e^{5y}$ is a function of $y$ , and $y$ is a function of $x$ , we apply the chain rule as follows: $(\frac{d}{dx})(e^{5y}) = (\frac{d}{dy})(e^{5y}) \cdot (\frac{dy}{dx})$ .
Differentiate with respect to $y$ : Differentiate $e^{5y}$ with respect to $y$ to get $5e^{5y}$ . So, $\frac{d}{dx}(e^{5y}) = 5e^{5y} \cdot \frac{dy}{dx}$ .
Substitute given derivative: Substitute the given $(\frac{dy}{dx}) = e^{5y}$ into the expression. We get $(\frac{d}{dx})(e^{5y}) = 5e^{5y} \cdot e^{5y}$ .
Simplify the expression: Simplify the expression by combining the exponential terms. We have $(d^{2}y)/(dx^{2}) = 5e^{5y} \cdot e^{5y} = 5e^{10y}$ .