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

Question

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

f(x) = (1+x²)^sin x

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says use logarithmic differentiation to find the derivative of the function F of X, which is equal to the quantity of 2 + X cubed in quantity raised to the cosine of X. Now we're asked to find the derivative of our function F of X using logarithmic differentiation. So our first step is to take a natural log of our function F of X. So we'll have the natural log of F of X. And that's going to be equal to the natural log of the quantity of 2 plus X cubed. In quantity, raised to the cosine of X. So we can use our properties of logarithms to rewrite the right hand side, mainly, we'll be using our power rule for logarithms. So we'll have the natural log of F of X. is equal to cosine of X, multiplied by the natural log of the quantity of 2 plus X Q in quantity. So now we need to take a derivative with respect to X of both sides of this equation. So, on the left hand side, we will have the derivative with respect to X of the natural log of F of X. So recall that when you take the derivative of your natural log function, you get 1 divided by the argument. So we'll we'll have here 1 divided by F of X. And that gets multiplied by the derivative of our argument. And so here we'll have the derivative of F with respect to X, which is going to be F of X. And this is going to be equal to. On the right hand side, we're going to need to use our product rule. So we call for the product rule that you're going to have the derivative of your first term in your product, which in this case the first term is going to be cosine of x. I want to take a derivative with respect to X of cosine of X, I will get minus sine of X, and then that gets multiplied by our second term in our product, which is the natural log of the quantity of 2 + x cubed in quantity. Then we'll have less, the first term in our product, which is cosine of X. Multiplied by the derivative of the 2nd term in our product. So here we'll have the derivative with respect to X of the natural log of the quantity of 2 + X cubed. So again, when we take the derivative of the natural log function, we get 1 divided by the argument, so we'll have 1 divided by. 2 plus execute. And that quantity gets multiplied by the derivative of our argument, so we'll have the derivative with respect to X of the quantity of 2 + X cubed, and that derivative is equal to 3X squared. So now we just need to simplify our expression on the right hand side. To do that, we'll need to write everything under a common denominator. So, we will have 1 divided by F of X. Multiplied by F of X. And that's going to be equal to, we'll have 3 X squared, multiplied by cosine of X. Minus the quantity of 2 + X cubed in quantity multiplied by sin of X. Multiplied by the natural log of the quantity of 2 + X cubed in quantity, and that whole quantity is divided by the quantity of 2 + X cubed. So now we need to solve this equation for FX. To do that, we need to multiply both sides by F of X. So we'll get F of X. is equal to F of X. Multiplying the quantity of 3 X squad, multiplied by cosine of X. Minus quantity of 2 + x cubed. In quantity multiplied by sin of X, multiplied by the natural log of the quantity of 2 + X cubed in quantity, and that whole quantity is still being divided by the quantity of 2 + X cubed. So now we need to substitute in our original expression for F of X, so we'll have F of X. is equal to the quantity of 2 plus X cubed. In quantity, raised to the cosine of X. And this gets multiplied by the quantity of 3 X squad, multiplied by cosine of X. Minus the quantity of 2 + X cubed. In quantity multiplied by sin of X. Multiplied by the natural log of the quantity of 2 + Xcute in quantity. This whole quantity is still being divided by the quantity of 2 + Xcut. So now we just need to simplify this expression, and so we will have F of X. Is going to be equal to the quantity of 2 plus X cubed. In quantity, rates to the quantity of cosine of X minus 1 in quantity, and this is multiplied by the quantity of 3 X squared multiplied by cosine of X. Minus the quantity of 2 + X cubed in quantity multiplied by sine of X, multiplied by the natural log of the quantity of 2 + X cubed in quantity. So this here is the final answer for this problem. So what we've done is we have found our derivative of our function F of X using logarithmic differentiation. 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) = (1+x²)^sin x

Key Concepts

Logarithmic Differentiation

Chain Rule

Implicit Differentiation

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