47–56. Derivatives of inverse functions at a point Consider th...

Question

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x)=4e^10x; (4,0)

Accepted Answer

Hello, in this video, we are going to be determining the value of an inverse derivative. So we are told that for the function, JFX is equal to 11, multiplied by E raise the power of 7 X, we want to determine the derivative of the inverse function at the point 11.0 without calculating its inverse. So we are given. J X equal to 11 multiplied by E raises the power of 7 X. And we are given the point 11.0. How can we determine the derivative of the inverse function without calculating the inverse? Well we actually have an equation that can help us do that. That equation is given to us as J inverse of B. is equal to 1 divided by J A. This is a bit of a confusing equation to understand, but what this is saying is that if you want to determine the value of the derivative of an inverse function, then that value is equal to one divided by J of the input value A. Now, how first we need to determine what our A and B values are going to be. Since we are working with an inverse function, recall that every output becomes inputs and vice versa. So that means that we are going to change the 0.11.0 to 0.11 because we are working with J inverse. And this means that the input value for our problem is going to be A is equal to 0. So now that we have the input value of A, we need to first find the derivative of our J function. So, in order to take the derivative of the given JFX function, we are going to use the exponential rule. J X will equal to 11 multiplied by the exponential role of E to the 7 X. That is going to be the original argument. E raised to the power of 7 X, multiplied by the natural logarithm of the base, which is natural logarithm of E, multiplied by the derivative with respect to X of the exponential function 7 X. So there's a bit of simplification we need to do for the for the derivative. First, the natural logarithm of E will simplify to be 1. And by using the power rule, the derivative with respect to X of 7 X is just going to be 7. 7 multiplied by 1 is just 7, and 7 multiplied by E erase the power of 7 X is just 7 multiplied by E erase the power of 7 X. So the derivative simplifies to 11 multiplied by 7, multiplied by E raises the power of 7 X, and if we multiply the coefficients together, that will give us a final derivative of 77 E raise the power of 7 X. This is going to be the derivative J X. Now, we need to use the identity for the value of the inverse derivative. Now what we need to do is we need to take the value of A, which we determined earlier, and plug it into the derivative. That is going to give us J A, which is 0 equal to 77 E raises the power of 7 multiplied by 0. This will simplify to 77 multiplied by Ease the power of zero, and anything with an exponential value of 0 is just 1. So this will simplify to be 77. So we have J A, which is 0 equal to 77. If we apply this to the identity, then the value of the derivative of the inverse, which is J inverse of B, which is going to be 11, will equal to 1 divided by 77. And this is going to be the solution to the problem. And the overall solution is going to be option C. So, I hope this video helps you in understanding how to approach this kind of problem, and I will go ahead and see you all in the next video.

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=4e^10x; (4,0)

Key Concepts

Inverse Functions

Derivative of Inverse Functions

Exponential Functions and Their Derivatives

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