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Find the derivative of the following function.

y=7^(6x^(6)-5x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=7^{6 x^{6}-5 x^{5}}$ $\newline$ 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^{6 x^{6}-5 x^{5}}

\newline

Answer:

y^{\prime}=

Identify function components: Identify the function and its components. $\newline$ The function is $y = 7^{6x^6 - 5x^5}$ , which is an exponential function with a base of $7$ and an exponent of $6x^6 - 5x^5$ .
Use chain rule for differentiation: Recognize that to differentiate $y = 7^{6x^6 - 5x^5}$ , we need to use the chain rule for exponential functions. $\newline$ The chain rule states that the derivative of $a^u$ , where $a$ is a constant and $u$ is a function of $x$ , is $a^u \cdot \ln(a) \cdot u'$ .
Differentiate exponent function: Differentiate the exponent function $u(x) = 6x^6 - 5x^5$ . To find $u'(x)$ , we apply the power rule to each term. $u'(x) = \frac{d}{dx}(6x^6) - \frac{d}{dx}(5x^5)$ $u'(x) = 36x^5 - 25x^4$
Apply chain rule to find derivative: Apply the chain rule to find the derivative of $y$ . Using the chain rule, we get $y' = 7^{(6x^6 - 5x^5)} \cdot \ln(7) \cdot (36x^5 - 25x^4)$ .
Simplify derivative expression: Simplify the expression for the derivative. $\newline$ $y' = (7^{6x^6 - 5x^5}) \cdot \ln(7) \cdot (36x^5 - 25x^4)$ $\newline$ This is the final form of the derivative.

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