 find an approximation of the equation of the tangent line of the given function at the given x-value .leave your answer in the form y=mx+c ,with m and correct to at least decimal places. y =tan(e1\4x2 cos (x2-2x)),at x=1

To find an approximation of the equation of the tangent line to the function at a given \(x\) value, we need to determine the slope of the tangent line at that point. This involves finding the derivative of the function \(y = \tan\left(e^{\frac{1}{4}x^2} \cos(x^2 - 2x)\right)\), evaluating it at \(x = 1\), and then using the point-slope form of the equation of a line to find the tangent line equation.

### Step 1: Differentiate the Function

Given the function:
\[
y = \tan\left(e^{\frac{1}{4}x^2} \cos(x^2 - 2x)\right)
\]

To find the derivative \( \frac{dy}{dx} \), we use the chain rule. If \(u = e^{\frac{1}{4}x^2} \cos(x^2 - 2x)\), then:
\[
y = \tan(u) \implies \frac{dy}{dx} = \sec^2(u) \cdot \frac{du}{dx}
\]

So, we first need to find \( \frac{du}{dx} \).

\[
u = e^{\frac{1}{4}x^2} \cos(x^2 - 2x)
\]

Using the product rule:

\[
\frac{du}{dx} = e^{\frac{1}{4}x^2} \frac{d}{dx}[\cos(x^2 - 2x)] + \cos(x^2 - 2x) \frac{d}{dx}\left[e^{\frac{1}{4}x^2}\right]
\]

### Step 2: Evaluate the Derivative at \(x = 1\)

Now, let's compute the derivatives step-by-step and evaluate them at \(x = 1\). I'll calculate this using Python to ensure precision.The slope of the tangent line at \(x = 1\) is approximately \(m = 0.5868\) (rounded to four decimal places). The value of \(y\) at \(x = 1\) is approximately \(y(1) = 0.8317\).

### Step 3: Find the Equation of the Tangent Line

The equation of the tangent line is given by:

\[
y - y_1 = m(x - x_1)
\]

Substitute \(x_1 = 1\), \(y_1 = 0.8317\), and \(m = 0.5868\):

\[
y - 0.8317 = 0.5868(x - 1)
\]

Simplify this equation to get it in the form \(y = mx + c\):

\[
y = 0.5868x + (0.8317 - 0.5868) \approx 0.5868x + 0.2449
\]

### Final Answer

The equation of the tangent line is:

\[
y = 0.5868x + 0.2449
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you find the derivative of a product of two functions?
2. What is the chain rule and how is it used in differentiation?
3. How can you use implicit differentiation to find the derivative of inverse trigonometric functions?
4. How do you determine concavity and inflection points from the second derivative?
5. What is the significance of the tangent line in graphing functions?

### Tip:
When working with trigonometric functions, remember that the derivative of \(\tan(x)\) is \(\sec^2(x)\), which can significantly affect the shape of the graph when calculating tangent lines.

Find an approximation of the equation of the tangent line of the given function at the given x-value. Leave your answer in the form y=mx+c, with m and c correct to at least 4 decimal places. y = tan(e^(1/4 * x^2) * cos(x^2 - 2x)), at x = 1

Learn how to find an approximation of the tangent line to the function y = tan(e^(1/4 * x^2) * cos(x^2 - 2x)) at x = 1. This solution uses calculus concepts like differentiation, the chain rule, and the product rule. Suitable for advanced high school or college-level students.

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