Differentiate cosh x cos x + sinh x sin x with respect to x
Understand the Problem
The question is asking to find the derivative of the given function, which involves differentiating a combination of hyperbolic and trigonometric functions with respect to x.
Answer
The derivative of the function is $f'(x) = \cosh(x) - \sin(x)$.
Answer for screen readers
The derivative of the function is given by $f'(x) = \cosh(x) - \sin(x)$.
Steps to Solve
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Identify the Function
We'll differentiate the given function, which is a combination of hyperbolic and trigonometric functions. -
Apply the Derivative Rules
Use the chain rule and the respective derivatives:
- The derivative of $\sinh(x)$ is $\cosh(x)$.
- The derivative of $\cos(x)$ is $-\sin(x)$.
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Differentiate Each Component
If our function is of the form $f(x) = \sinh(x) + \cos(x)$, the derivative will be: $$ f'(x) = \frac{d}{dx}[\sinh(x)] + \frac{d}{dx}[\cos(x)] $$ -
Calculate the Derivative
Substituting the derivatives from step 2: $$ f'(x) = \cosh(x) - \sin(x) $$ -
Write the Final Result
Make sure to present the result clearly: The derivative of the function is: $$ f'(x) = \cosh(x) - \sin(x) $$
The derivative of the function is given by $f'(x) = \cosh(x) - \sin(x)$.
More Information
The derivative provides the rate at which the function changes with respect to $x$. The function consists of both hyperbolic and trigonometric components, making the calculation unique and showcasing the use of different derivative rules.
Tips
- Forgetting the negative sign when differentiating $\cos(x)$.
- Confusing hyperbolic functions with corresponding trigonometric functions.
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