"In Exercises 59–86, find the derivative of y with respect to ...

Question

"In Exercises 59–86, find the derivative of y with respect to the given independent variable.

67. y = 7^(sec θ) ln 7"

Accepted Answer

Welcome back, everyone. Compute D Y divided by DX4 Y equals 5 to the power of sin x multiplied by LN of 5. So for this problem, we want to find the derivative of Y. Notice that LA of 5 is a constant, so we can factor out LN of 5 and multiply it by the derivative of 5 to the power of sine of X. Now let's notice that 5 to the power of sin of X is a composite function. So we can express 5 to the power of sine of x as 5 to the power of U, where u is sin of X, right? So now the derivative of 5 power of sin of X is going to be found by using the chain rule. Let's remember that first of all, we are differentiating. Our function with respect to sin of X, our exponent, right? Because essentially, according to the gene rule, We can basically write this derivative as. D Y divided by DU multiplied by DU divided by D X. And we can basically assign 5 power of X X equals 5 power of U, which is equal to Y, right? So now we want to differentiate why with respect to you. Which is the derivative of 5 power of U, and this is 5 power of U multiplied by LN of 5. We can immediately express it in terms of X, because U is sin of X. We get 5 to the power of sin of X multiplied by LN of 5. And now we also want to find the derivative of you with respect to X. U is equal to sin of x, so we want to find the derivative of sine, which is cosine. And now let's go ahead and combine. So the derivative of 5 to the power of sin of X is going to be equal to 5 power of sin of X, multiplied by LN 5 multiplied by cosine of X. And now going back into the derivative of the original function, the whole function Y, including the constant LN of 5, we simply want to multiply our result by LN of 5. So LN of 5 is going to get squared. So the final answer is going to be 5 to the power of sin of X. Multiplied by lnquad of 5 multiplied by cosine of X, and that's our final answer for this problem. Thank you for watching.

"In Exercises 59–86, find the derivative of y with respect to the given independent variable.67. y = 7^(sec θ) ln 7"

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"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"