Theorem 7.8
Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) ...

Question

Theorem 7.8

Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).

Accepted Answer

Welcome back, everyone. Given the inverts of cash of U equals LN of U + square root of U2 minus 14 U greater than 1, find the derivative of inverts of cash of 3 X. For this problem, let's begin by evaluating the derivative of inverse of cache of u. What we want to do is simply differentiate LN of U + square root of u2 minus 1, and we can do that by applying the chain rule, right? So to begin with, we're differentiating LN of U plus square root of u2 minus 1 with respect to the inner function. We get a 1 divided by the inner function. Or basically one divided by u plus square root of u2 minus 1, and according to the chain rule we have to multiply by the derivative of the inner function. So we are multiplying by the derivative of u plus square root of u2 minus 1. Let's go ahead and simplify, so we have 1 divided by U plus square root of U2 minus 1 multiplied by. The derivative of u is 1 plus the derivative of the radical term can be obtained by differentiating u2 minus 1 raises the power of 1/2, right. And then we can apply the power rule. We get 1/2 multiplied by u2 minus 1 to the power of -12. And according to the chain rule for this function, we're multiplying by the derivative of u2 minus 1. So now let's identify the final derivative and simplify. This is equal to 1 divided by U plus square root of u2 minus 1. Multiplied by one plus. 1 divided by 2 square root of U2 minus 1 multiplied by 2 U, right, because the derivative of U2 minus 1 is 2 U. We call that the derivative of a constant is 0. So we can write to you in the numerator, and we can clearly see that we can cancel out. The constant too. And now from here we can simplify. We can find the common denominator for the second factor, and we get 1 divided by U + square root of U2 minus 1 for the first one. Or the second factor, we can write the common denominator, which is square root of U2 minus 1. And our numerator is going to have square root of U 2 minus 1 plus U. Notice that the opposite terms can now be canceled out and we get the derivative of one divided by. Square root of u squad minus 1. So we have the general expression for the derivative for the inverse of cache of you. Now if we want to evaluate the derivative of inverse of cash of 3 X. We can basically say that it's going to be one divided by square root of. U 2 minus 1, so in this problem U is equal to 3X, meaning we get 3 X2 minus 1. And because EU is now a function of X, we're applying the chain rule and multiplying by the derivative of 3 X. All that we have to do is simplify. We get The derivative of 3 X is 3, so we get 3 in the numerator and our denominator. It's going to contain 9 x 2 minus 1. So that will be our final answer. Let's label it, and thank you for watching.

Theorem 7.8
Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).

Key Concepts

Inverse Hyperbolic Sine Function (sinh⁻¹ x)

Differentiation of Logarithmic Functions

Chain Rule and Derivative of Composite Functions

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