Differentiate x^cosh(x) w.r.t. x
Understand the Problem
The question is asking to find the derivative of the function x raised to the hyperbolic cosine of x with respect to x. This involves applying differentiation rules, possibly including the chain rule and product rule.
Answer
$$ \frac{dy}{dx} = x^{\cosh(x)} \left( \sinh(x) \ln(x) + \frac{\cosh(x)}{x} \right) $$
Answer for screen readers
The derivative of the function $y = x^{\cosh(x)}$ with respect to $x$ is
$$ \frac{dy}{dx} = x^{\cosh(x)} \left( \sinh(x) \ln(x) + \frac{\cosh(x)}{x} \right) $$
Steps to Solve
- Write the Function Clearly
The function we need to differentiate is written as:
$$ y = x^{\cosh(x)} $$
- Apply the Natural Logarithm
To differentiate, we take the natural logarithm of both sides to simplify the expression:
$$ \ln(y) = \ln(x^{\cosh(x)}) $$
Using the property of logarithms that $\ln(a^b) = b\ln(a)$, we rewrite it as:
$$ \ln(y) = \cosh(x) \ln(x) $$
- Differentiate Both Sides
Differentiate both sides with respect to $x$. We will use the chain rule on the left side and the product rule on the right side:
$$ \frac{1}{y}\frac{dy}{dx} = \frac{d}{dx} \left( \cosh(x) \ln(x) \right) $$
Differentiating the right side gives:
$$ \frac{d}{dx} \left( \cosh(x) \ln(x) \right) = \sinh(x) \ln(x) + \frac{\cosh(x)}{x} $$
- Solve for $\frac{dy}{dx}$
Now, multiply both sides by $y$ to solve for $\frac{dy}{dx}$:
$$ \frac{dy}{dx} = y \left( \sinh(x) \ln(x) + \frac{\cosh(x)}{x} \right) $$
- Substitute Back for y
Finally, substitute $y$ back in for $x^{\cosh(x)}$:
$$ \frac{dy}{dx} = x^{\cosh(x)} \left( \sinh(x) \ln(x) + \frac{\cosh(x)}{x} \right) $$
The derivative of the function $y = x^{\cosh(x)}$ with respect to $x$ is
$$ \frac{dy}{dx} = x^{\cosh(x)} \left( \sinh(x) \ln(x) + \frac{\cosh(x)}{x} \right) $$
More Information
This derivative is useful in various applications, such as in physics and engineering, where hyperbolic functions arise in scenarios involving exponential growth or decay. Understanding how to differentiate complex functions is fundamental in calculus.
Tips
- Failing to properly apply the product rule and chain rule when differentiating the logarithmic form can lead to incorrect results.
- Not simplifying the logarithmic expression sufficiently before differentiating.
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