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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))=◻" I " bar(+)

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 $y$ . $\newline$ $\frac{d^{2} y}{d x^{2}}=\square \text { I } \overline{+}$

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Find the vertex of the transformed function

Full solution

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}}=\square \text { I } \overline{+}

Given first 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 $x$ . The chain rule states that the derivative of $e^{u}$ with respect to $x$ is $e^{u} \cdot \frac{du}{dx}$ , where $u$ is a function of $x$ . In this case, $u = 5y$ , so we need to multiply $e^{5y}$ by the derivative of $5y$ with respect to $x$ , which is $x$ $1$ .
Substitute into expression: We already know that $(\frac{dy}{dx}) = e^{5y}$ , so we substitute this into our expression for the derivative of $5y$ with respect to $x$ . This gives us $5 \cdot e^{5y}$ .
Find second derivative: Therefore, the second derivative $\frac{d^2y}{dx^2}$ is $e^{5y} \cdot 5 \cdot e^{5y}$ , which simplifies to $5 \cdot e^{10y}$ .

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