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Find the derivative of the following function.

y=7^(x^(2)-9x)
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=7^{x^{2}-9 x}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=7^{x^{2}-9 x}

\newline

Answer:

y^{\prime}=

Identify Function Components: Identify the function and its components. $\newline$ We have $y = 7^{(x^2 - 9x)}$ , which is an exponential function with base $7$ and an exponent that is a function of $x$ , namely $x^2 - 9x$ .
Apply Chain Rule: Apply the chain rule for differentiation. $\newline$ 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. $\newline$ Here, the outer function is $a^u$ where $a$ is a constant and $u$ is a function of $x$ . The derivative of $a^u$ with respect to $u$ is $a^u \cdot \ln(a)$ . $\newline$ The inner function is $u(x) = x^2 - 9x$ . The derivative of $u$ with respect to $x$ is $a$ $0$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ Let's denote the outer function as $f(u) = 7^u$ . Then, $f'(u) = 7^u \cdot \ln(7)$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . We have $u(x) = x^2 - 9x$ . The derivative $u'(x)$ is found by differentiating each term separately: $u'(x) = \frac{d}{dx}(x^2) - \frac{d}{dx}(9x)$ $u'(x) = 2x - 9$
Apply Chain Rule for Derivative: Apply the chain rule to find the derivative of $y$ with respect to $x$ . Using the chain rule, we get: $y'(x) = f'(u(x)) \cdot u'(x)$ $y'(x) = (7^{x^2 - 9x} \cdot \ln(7)) \cdot (2x - 9)$
Simplify Derivative: Simplify the expression for the derivative. $\newline$ $y'(x) = 7^{(x^2 - 9x)} \cdot \ln(7) \cdot (2x - 9)$ $\newline$ This is the final form of the derivative.

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