Find an equation of the tangent line at the point indicated....

Key Concepts

Tangent Line

A tangent line to a function at a given point is a straight line that touches the graph of the function at that point and has the same slope as the function does at that point. It provides a linear approximation of the function near the point of tangency, and its equation is typically written using the point-slope form.

Derivative

The derivative of a function measures the instantaneous rate of change of that function with respect to its variable. It is used to determine the slope of the tangent line at any given point on the function.

Exponential Function Differentiation

When differentiating an exponential function of the form a^(u(x)), where a is a positive constant and u(x) is a differentiable function, the derivative is found using the formula: a^(u(x)) * ln(a) * u'(x). This captures how the exponential function changes relative to changes in the exponent.

Chain Rule

The chain rule is a technique for differentiating composite functions. It states that if a function can be expressed as a composition of two or more functions, then its derivative is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. This principle is essential when the exponent itself is a function of the variable.

Transcript

00:01 So we have the following function, and we want to find the equation of the tangent line to this function at x equals 1.

00:09 So we need a couple things.

00:10 Our goal is to find a line, right? we want to find something that looks like y equals mx plus b.

00:17 And the way we can find m is to take the derivative at this point for x.

00:24 So in other words, we want to find y prime evaluated at x equals 1.

00:29 So first we need to find our derivative.

00:32 So let's find y prime.

00:34 So we've got an exponential here.

00:36 So we know the derivative of an exponential like this is equal to the exponential itself.

00:44 So pi to the 5x minus 2.

00:47 But we need to do something.

00:48 We need to multiply by ln of the base.

00:52 So ln of pi gets multiplied by.

00:56 And then we have the chain roll here.

01:03 And the chain roll.

01:04 Is that we need to multiply by the derivative of the power.

01:08 So derivative of 5x minus 2.

01:12 So let's see what we got here.

01:13 We've got pi to the 5x minus 2 times ln pi times the derivative of this guy is just 5.

01:26 So let's see ln pi times 5.

01:35 So this part here, and we're going to do some rounding here is approximately 5 .72.

01:46 So if we first we define our derivative y prime evaluated at x equal to one we've got pi raised to be if x is one this is going to be five minus two which is the third power and this gets multiplied by this 5 .72 whatever so let's see here doing this gives us approximately 177 .5 if we round to just one decimal place so this guy is our slope...

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