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

Question

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

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Identify Functions: Identify the outer function and the inner function for the chain rule. 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)) * h'(x).Differentiate Outer Function: Differentiate the outer function with respect to the inner function. 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. 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)) * 7x^(6).Write Final Answer: Write the final answer using symbolic notation. f'(x) = 7x^(6) * cosh(x^(7)).

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

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Calculate the derivative of \newlinef(x)=sinh⁡(x7).f(x)=\sinh(x^{7}).f(x)=sinh(x7).\newline(Use symbolic notation and fractions where needed.)

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