Differentiate a^sinh(2x) with respect to x.
Understand the Problem
The question is asking us to find the derivative of the function a^sinh(2x) with respect to x. This involves using differentiation rules, including the chain rule and properties of hyperbolic functions.
Answer
$$ \frac{dy}{dx} = 2 \ln(a) a^{\sinh(2x)} \cosh(2x) $$
Answer for screen readers
The derivative of the function $a^{\sinh(2x)}$ with respect to $x$ is: $$ \frac{dy}{dx} = 2 \ln(a) a^{\sinh(2x)} \cosh(2x) $$
Steps to Solve
- Identify the function to differentiate
We have the function $a^{\sinh(2x)}$. To differentiate this, we will need to use the chain rule and properties of exponential and hyperbolic functions.
- Rewrite using logarithmic differentiation
Using the property of logarithms, we can rewrite the function to facilitate differentiation: $$ y = a^{\sinh(2x)} $$ Taking natural logs gives: $$ \ln(y) = \sinh(2x) \ln(a) $$
- Differentiate both sides
Now, differentiate both sides with respect to $x$. Remember that we will apply the chain rule on the left side: $$ \frac{1}{y} \frac{dy}{dx} = \cosh(2x) \cdot 2 \ln(a) $$
- Isolate $\frac{dy}{dx}$
Multiply both sides by $y$ to isolate $\frac{dy}{dx}$: $$ \frac{dy}{dx} = y \cdot \cosh(2x) \cdot 2 \ln(a) $$
- Substitute back for $y$
We know $y = a^{\sinh(2x)}$, so substitute this back in: $$ \frac{dy}{dx} = a^{\sinh(2x)} \cdot \cosh(2x) \cdot 2 \ln(a) $$
- Final Form of the Derivative
Thus, the derivative of the function is: $$ \frac{dy}{dx} = 2 \ln(a) a^{\sinh(2x)} \cosh(2x) $$
The derivative of the function $a^{\sinh(2x)}$ with respect to $x$ is: $$ \frac{dy}{dx} = 2 \ln(a) a^{\sinh(2x)} \cosh(2x) $$
More Information
This derivative involves the use of hyperbolic functions and demonstrates how logarithmic differentiation can simplify finding derivatives of exponential functions with complicated exponents.
Tips
- Forgetting the chain rule: Make sure to always apply the chain rule correctly when differentiating functions with compositions.
- Misapplying hyperbolic identities: Ensure that the identities for hyperbolic functions, such as $\sinh$ and $\cosh$, are applied correctly.
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