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Find the equation of the line tangent to $y=2^{\sin x}$ at $x=\pi / 2 .$ Graph the function and the tangent line.

y = 2

Calculus 1 / AB

Chapter 3

Derivatives

Section 9

Derivatives of Logarithmic and Exponential Functions

Differentiation

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the equation of why equal to Mari equal to sign Moved to the sign of X. Excuse me. I need to find the tangent line the point X equal to hi over to. So to find the tangent line, we first need to find the point that the line we'll cross work, meet with the initial graph. So we need to plug in hi over to to our equation. So sign of pi over two so we can get that to to the sign Hi were to It's going to equal sign up high over two is One said this goes to a one equals two to the one which is equal to two So we need our tangent line to cross the point Hi or two to now we need to find the rate of change at that current point So we need to take the derivative. So to do this we need to use algorithmic differentiation. So we get natural log of y equal to sign of X natural log to take the derivative of both sides and we get why prime over Why? Because the derivative of natural walk. Why is why times the derivative of why my prime is equal to the derivative of sign of X times The natural look too. And the derivative sign of X is co sign axe Times match along too. Now we need to multiply both sides by Why So we get why prime equal to co sign X natural log of to times two to sign X Now we can plug in hi over to at in this equation because we have now found a derivative. And we can quickly do this because co sign it was a horrible see oh, sign hi over to zero. So we know that since that is equal to zero, we get zero times the natural water, too times to to the sign Hi over two. But since we have a zero multiplied by everything, this goes to zero. And when we have a one y prime equal to zero, that means our tangent line going to be horizontal. So are tangent line. We're going to call this line t equal to using the equation and X plus be and I m is 00 and be equal to two as we solved above. So we get t is equal to to and that is the equation for a tangent line. But the pro also says we need to graft this. So I am going to drag in window for graphing calculator and so we can see quickly are graph of to Science X already And now I can turn on the plot for our tangent line. Why is equal to two? We can see that it just crosses and meats our initial graph, you know, just a little. And there we go. You need to plot with a raft calculator. Copy down this graph to to the sine x No, I equal to two. And this is Dismas is a free online graphing calculator for anyone to use as well. But I am going to switch back to our solution here. Zoom helps for all of the work. There we are.

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The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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