Find the Derivative - d/dx y=(e^(7x)+e^(-7x))/x

Problem

d()/d(x)(e(7*x)+e(−7*x))/x

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two expressions, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions. Let u=e(7*x)+e(−7*x) and v=x

3. Differentiate the numerator u using the chain rule.

d(u)/d(x)=7*e(7*x)−7*e(−7*x)

4. Differentiate the denominator v

d(v)/d(x)=1

5. Substitute these components into the quotient rule formula.

d(y)/d(x)=(x*(7*e(7*x)−7*e(−7*x))−(e(7*x)+e(−7*x))*(1))/(x2)

6. Simplify the numerator by distributing x and removing the parentheses.

d(y)/d(x)=(7*x*e(7*x)−7*x*e(−7*x)−e(7*x)−e(−7*x))/(x2)

Final Answer

d()/d(x)(e(7*x)+e(−7*x))/x=(7*x*e(7*x)−7*x*e(−7*x)−e(7*x)−e(−7*x))/(x2)

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