75–86. Logarithmic differentiation Use logarithmic differentia...

Question

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = tan¹⁰x / (5x+3)⁶

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to use logarithmic differentiation to find the derivative of F of X, which is equal to sin to the 7th of X, divided by the quantity of 3 X2 + 2 in quantity rates to the 4th power. We give them 4 possible choices as our answers. For choice A, we have F of X is equal to minus the 7th of X, multiplied by the quantity of 21 X cubed multiplied by cotangent of X, plus 14 multiplied by tangent of X, plus 24 X in quantity divided by the quantity of 3 X plus 2 in quantity raised to the 5th. For choice B, we have F of X is equal to 7 multiplied by the 6th of X, multiplied by cosine of X, divided by the quantity of 12 multiplied by the quantity of 3 X plus 2 in quantity cubed. For choice C, we have F of X is equal to 7 multiplied by the sin to the 6th of X, divided by the quantity of 4 multiplied by the quantity of 3 x plus 2 in quantity cubed. And for choice d, we have F of X is equal to the 7th of X multiplied by the quantity of 21 x2 multiplied by the cotangent of X, plus 14 multiplied by the cotangent of X minus 24 X in quantity divided by the quantity of 3 X2 + 2 in quantity rates to the fifth. Now, this question was to define the derivative of F of X using logarithmic differentiation. So our first step is to take the natural log of our function F of X. So we'll have the natural log of F of X. is equal to the natural log of the quantity of sin to the 7th of X divided by the quantity of 3 x 2 plus 2 in quantity raised to the 4th. So we can rewrite the right hand side using our properties of logarithms, namely the power rule and quotient rule. So doing so, we will have the natural log of F of X. is equal to 7 multiplied by the natural log of sine of X. Minus 4 multiplied by the natural log of the quantity of 3 X2 + 2 in quantity. And now we need to take a derivative with respect to X of both sides of this equation. In doing so on the left hand side, we will have the derivative with respect to X of the natural log of F of X. And recall that when you take the derivative of the natural law function, you will get 1 divided by the argument. So I'll have 1 divided by F of X. But we also need to apply our chain rule, so we'll need to multiply this by the derivative of the argument. So we'll have the derivative of F with respect to X, which is F of X. And this is going to be equal to, on the right hand side, I'll have 7, multiplied by the derivative with respect to X of the natural log of the sin of X. And so again, when we take the derivative of the natural law, we'll get 1 divided by the argument, so we'll have 7 multiplied by 1 divided by sine of X. We'll need to also multiply this by the derivative of the argument, so I'll have the derivative with respect to X of sin of X, which is cosine of X. Then we'll have minus 4 multiplied by the derivative with respect to X of the natural log of the quantity of 3 X2 + 2. Again, taking the derivative of the natural log function, I will get 1 divided by 3 X2 + 2. In quantity, and that gets multiplied by the derivative with respect to X of 3 X2 + 2, which is 6 X. So now we just need to simplify our right hand side of our expression here. So we'll have 1 divided by F of X. Multiplied by F of X. is equal to 7 multiplied by the cotangent of X. Since you, if you recall that cosine of X divided by sin of X is cotangent of X, then we'll have minus 24 X divided by the quantity of 3 X squared plus 2. Now, we want to write everything on the right hand side over uh a common denominator. So doing so, we'll have one divided by F of X. Multiplied by F of X. is equal to 7 multiplied by cotangent of X, multiplied by the quantity of 3 X 2 plus 2 in quantity. Minus 24 X. And that whole quantity is divided by the quantity of 3 X2 + 2. So now we need to solve this equation for F of X. To do that, we'll just multiply both sides of the equation by F of X. So we'll have F of X. is equal to F of X. Multiplying the quantity. And here we'll simplify our expression that we originally had on the right-hand side. So we'll distribute the seven cotangent of X, and so we'll have 21 X squared multiplied by cotangent of X, plus 14 multiplied by the cotangent of X. Minus 24 X, that whole quantity is divided by. The quantity of 3 X2 + 2 in quantity. So now we need to substitute in our original expression for F of X. Doing so, we'll have F of X. is equal to Assigned to the 7th of X, divided by the quantity of 3 x 2 plus 2 in quantity rates to the 4th, that fraction is multiplied by the quantity of 21 X2, multiplied by cotangent of X. Plus 14, multiplied by the cotangent of X minus 24 X in quantity, divided by the quantity of 3 X 2 plus 2 in quantity. So now we just need to simplify this this expression, so we'll have F of X. It's equal to Assigned to the 7th of X, multiplying the quantity of 21 X squad, multiplied by cotangent of X. Plus 14 multiplied by the cotangent of X minus 24 X in quantity, divided by the quantity of 3 x 2 plus 2 in quantity raised to the 5th power. So this here is the expression that we are actually looking for for this problem. And so now we just need to compare our answer to our answer choices. And in doing so, We see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = tan¹⁰x / (5x+3)⁶

Key Concepts

Logarithmic Differentiation

Chain Rule

Product and Quotient Rules

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