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Find the derivative of the following function.

y=5^(8x^(2))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=5^{8 x^{2}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=5^{8 x^{2}}

\newline

Answer:

y^{\prime}=

Identify Components: Identify the components of the function. $\newline$ The function $y = 5^{8x^{2}}$ is an exponential function where the base is a constant ( $5$ ) and the exponent is a function of $x$ ( $8x^{2}$ ).
Apply Chain Rule: Apply the chain rule for derivatives. $\newline$ The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $5^u$ and the inner function is $u = 8x^{2}$ .
Find Derivative Outer: Find the derivative of the outer function with respect to $u$ . The derivative of $a^u$ with respect to $u$ , where $a$ is a constant, is $a^u \cdot \ln(a)$ . Therefore, the derivative of $5^u$ with respect to $u$ is $5^u \cdot \ln(5)$ .
Find Derivative Inner: Find the derivative of the inner function with respect to $x$ . The inner function is $u = 8x^{2}$ . The derivative of $8x^{2}$ with respect to $x$ is $16x$ , since $\frac{d}{dx}(x^{2}) = 2x$ and we have a constant multiple of $8$ .
Apply Chain Rule: Apply the chain rule using the derivatives from steps $3$ and $4$ . $\newline$ The derivative of $y$ with respect to $x$ is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This gives us $y' = (5^{8x^{2}} \cdot \ln(5)) \cdot (16x)$ .
Simplify Derivative: Simplify the expression for the derivative. $\newline$ $y' = 16x \cdot 5^{8x^{2}} \cdot \ln(5)$ . This is the simplified form of the derivative of the given function.

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