Differentiate x sinh x with respect to x
Understand the Problem
The question is asking for the derivative of the function x * sinh(x) with respect to x. To solve this, we will apply the product rule of differentiation, which states that if we have two functions multiplied together, the derivative is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Answer
The derivative is $y' = x \cosh(x) + \sinh(x)$.
Answer for screen readers
The derivative of the function $x \cdot \sinh(x)$ with respect to $x$ is: $$ y' = x \cosh(x) + \sinh(x) $$
Steps to Solve
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Identify the functions We have two functions: Let $u = x$ and $v = \sinh(x)$.
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Differentiate the first function The derivative of $u = x$ with respect to $x$ is: $$ \frac{du}{dx} = 1 $$
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Differentiate the second function The derivative of $v = \sinh(x)$ with respect to $x$ is: $$ \frac{dv}{dx} = \cosh(x) $$
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Apply the product rule According to the product rule, the derivative $y' = \frac{d}{dx}(uv)$ is given by: $$ y' = u \frac{dv}{dx} + v \frac{du}{dx} $$ Substituting the values we found: $$ y' = x \cdot \cosh(x) + \sinh(x) \cdot 1 $$
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Combine the terms This simplifies to: $$ y' = x \cosh(x) + \sinh(x) $$
The derivative of the function $x \cdot \sinh(x)$ with respect to $x$ is: $$ y' = x \cosh(x) + \sinh(x) $$
More Information
The hyperbolic sine function, $\sinh(x)$, is similar to the ordinary sine function in trigonometric identities, but it describes the relationship to hyperbolas. Its derivative, $\cosh(x)$, is linked to many physical phenomena, including calculations in physics involving hyperbolic functions.
Tips
- Forgetting to apply the product rule correctly by missing one of the functions in differentiation.
- Confusing hyperbolic functions with their trigonometric counterparts, which can lead to incorrect derivatives.
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