Find an equation of the tangent line to the graph of y=x^x+2...

Key Concepts

Tangent Line

The tangent line to a function at a given point is the line that just touches the curve and has the same instantaneous slope at that point. It serves as the best local linear approximation for the function near that specific point.

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input. Evaluating the derivative at a particular point yields the slope of the tangent line at that point, which is essential for constructing the tangent line equation.

Logarithmic Differentiation

Logarithmic differentiation is a technique used for differentiating functions where variables appear in both the base and the exponent. By taking the natural logarithm of both sides of the equation, the problem is transformed into a form where the product and chain rules can be more easily applied.

Chain Rule

The chain rule is a key differentiation tool used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function, enabling the handling of nested functions during differentiation.

Transcript

00:01 We're going to be finding an equation for the tangent line to this function at x is equal to 1.

00:05 So we're going to have to find the derivative of this function, but first we're also going to have to find the y value that corresponds with this x value.

00:13 So let's go ahead and plug in x is equal to 1.

00:15 So we'd have y is equal to 1 to the 1 plus 2 power, which is just 1.

00:21 And so we're looking at the point 1 comma 1.

00:24 Now the next thing we can do is take the derivative of this function.

00:27 In order to take the derivative, we're going to first have to take the natural log, both sides.

00:30 So we're going to be using logarithmic differentiation.

00:34 So we have the natural log of y is equal to the natural log of x, the x plus two power.

00:39 We can take this x plus two and put it out front due to the properties of natural logs.

00:44 So we have the natural log of y is equal to x plus two times the natural log of x.

00:50 Now we can take the derivative of both sides.

00:53 On the left side, we're going to get y -prime divided by y, or one divided by y times y prime.

00:58 And on the left side, this is going to be a product rule derivative.

01:03 So we're going to have the derivative of x plus 2, which is just one times the natural log of x.

01:08 So it's just going to be the natural log of x, and then plus x plus 2 times the derivative of the natural log of x, which is 1 divided by x.

01:15 So we're going to have x plus 2 divided by x there...

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