Derivatives in Differential Form

Question

In Exercises 17–28, find dy.

y = sin(5√x)

Accepted Answer

Welcome back, everyone. Determine D Y for the following function Y equals cosine of 13 square root of X. For this problem, let's recall that the derivative of Y or Y is equal to D Y divided by DX right in a differential form. So if we want to solve for D Y, we can just rearrange it and we can show that D Y is simply the product between the derivative and D X. All that we have to do is identify Y, which is the derivative of cosine of 13 square root of X, and here we have a composite function. So we have to apply the chain rule. First of all, we're differentiating cosine with respect to. 13 square root of X. According to the table of basic derivatives, the derivative of cosine is negative sine. Our angle remains the same, which is 13 square root of X. But now, according to the chain rule, we have to multiply by the derivative of the inner function. So we're multiplying by the derivative of 13. Square root of X. This gives us negative signs. Of 13 square root of X multiplied by 13. Multiplied by the derivative of square root of X, which is the derivative of X-rays to the power of one half. And here we can apply the power rule which gives us negative sign. Of 13 square root of X multiplied by 13, multiplied by 1/2, X rates the power of 1/2 minus 1, which is negative 1/2. And now let's simplify. Now, first of all, we can take out 13 divided by 2 as a constant, and we also have a negative sign in front, we get -13 halves. And we also have a negative exponent for X-12, so we can write X with a positive exponent in the denominator, and because that exponent is 12, we can just rewrite square root of X instead of X the power of 1/2. So, first of all, we have -13 divided by 2 square root of X. And finally, we have to include the sign. So we get sign. Of 13 square root of x. Well done, so we have got our derivative, and that we can conclude that the expression becomes d y equals our derivative. 13 divided by 2 square roots of X. Sine of 13 square root of X we are done with the derivative and then we simply need to add the X to finish the expression. That's it. Thank you for watching.

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sin(5√x)

Key Concepts

Chain Rule

Derivative of Sine Function

Derivative of Square Root Function

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