Calculate the derivative of the following functions.
y =...

Question

Calculate the derivative of the following functions.

y = tan e^x

Accepted Answer

Welcome back, everyone. Find the derivative of the function y equals cotangent of 5 eats the power of X, and we're given 4 answer choices A says -1/5, E X C 25 eats the power of X. B says -5 eats. Power of X C 25 eats the power of X. C says 1/5 eats the power of X. 25 eats the power of X, and D says -5 eats the power of X. 25 e to the power of x. First of all, let's analyze the given function. It says Y equals. Cotangent of 5 e to the power of X. We have to understand that this is a composite function, so it consists of the outer function, which is cotangent, in the inner function which is 5 e to the power of X. And what we're going to do is simply set a variable for the inner function and rewrite the function as cotangent of u, where U is equal to 5 e to the power of x. And now if we want to differentiate. Why we're going to apply the chain rule because we have a composite function, and the chain rule says that the derivative is equal to Dy divided by the U, so essentially differentiating Y with respect to u, and then we're multiplying the derivative of u. Relative to X, right, so DU divided by D X. So we're going to apply the chain rule and we're going to begin from the very final part. We want to identify the U divided by the X. Essentially this is the derivative of U with respect to X. So let's evaluate the derivative of 5 e to the power of X. We have a constant so we can write 5, and the derivative of e to the power of X is just e to the power of X from the table of derivatives, right? It is the basic derivative. Then what we want to do is evaluate the Y D Y divided by the U, right? So the derivative of Y relative to U. So specifically we are calculating the derivative of cotangent. Ah for you Now, once again, this is a basic derivative, we have to know that this is negative cos 2 of you. And that's it. We have two parts that we need. Let's label them as part 1. This is D Y divided by D U and then part 2, which is DU divided by DX. And if we use Thus we can say that Y the derivative of Y is equal to negative cosequence 2 U. Now what is you? Well, it is 5. Multiplied by the power of X, and then we're multiplying by EU divided by D X which is 5 e to the power of x. And if we simply write the coefficient in front as well as E X, we get -5 E to the power of X multiplied by coc2 of 5 e to the power of X, which essentially corresponds to the answer choice d. Thank you for watching.

Calculate the derivative of the following functions.
y = tan e^x

Key Concepts

Derivative

Chain Rule

Trigonometric Functions

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