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Calculate the derivative of $\newline$ $f(x)=\sinh(x^{7}).$ $\newline$ (Use symbolic notation and fractions where needed.)

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Find derivatives of logarithmic functions

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Q. Calculate the derivative of

\newline

f(x)=\sinh(x^{7}).

\newline

(Use symbolic notation and fractions where needed.)

Identify Functions: Identify the outer function and the inner function for the chain rule. $\newline$ The outer function is $\sinh(u)$ , and the inner function is $u = x^{7}$ . We will apply the chain rule which states that if $f(x) = g(h(x))$ , then $f'(x) = g'(h(x)) \cdot h'(x)$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $\sinh(u)$ with respect to $u$ is $\cosh(u)$ . So, if we let $u = x^{7}$ , then the derivative of $\sinh(u)$ is $\cosh(x^{7})$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $x^{7}$ with respect to $x$ is $7x^{6}$ .
Apply Chain Rule: Apply the chain rule by multiplying the derivatives from Step $2$ and Step $3$ . $\newline$ The derivative of $f(x)$ with respect to $x$ is the product of the derivative of the outer function and the derivative of the inner function. Therefore, $f'(x) = \cosh(x^{7}) \times 7x^{6}$ .
Write Final Answer: Write the final answer using symbolic notation. $\newline$ $f'(x) = 7x^{6} \cdot \cosh(x^{7})$ .

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