Question

Find the slope of the line tangent to the graph of $y={\sin }^{-1}x$ at $x=0$.

Short answer

Answer: The slope of the tangent line at $x=0$ is 1.

Step-by-step solution

Differentiate the function

To differentiate $y={\sin }^{-1}x$ with respect to $x$, we can use the chain rule. We know that the derivative of ${\sin }^{-1}x$ is $\frac{1}{\sqrt{1-x^2}}$. So, we get the derivative as: $\frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}$

Find the slope at $x=0$

Now, substitute $x=0$ into the derivative expression: $\frac{dy}{dx}{|}_{x=0}=\frac{1}{\sqrt{1-0^2}}=\frac{1}{\sqrt{1}}=1$ So, the slope of the tangent line to the graph of $y={\sin }^{-1}x$ at $x=0$ is 1.

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the trigonometric functions such as sine, cosine, and tangent. These functions are essential in various branches of mathematics including calculus, geometry, and trigonometry.

When dealing with the function like the sine, for a given angle, it provides us with the ratio of the length of the opposite side to the hypotenuse in a right triangle. The inverse sine, denoted as ${\text{sin}}^{-1}\text{ or arcsin}$, does the reverse: for a given ratio value, it determines the angle that corresponds to it.

It is necessary to understand that inverse trig functions are functions that 'undo' the trigonometric functions. For example, if $\text{sin}(\theta )=x$, then $\theta ={\text{sin}}^{-1}(x)$. But they have restricted domains and ranges to ensure they are functions. For the inverse sine, typically the range is restricted to $-\frac{\pi }{2}$ to $\frac{\pi }{2}$, and the domain from -1 to 1.

Derivative of Inverse Sine

The derivative of the inverse sine function can be quite unintuitive when first encountered. It's a cornerstone in calculus that serves as a gateway to understanding change in trigonometric contexts.

To find the derivative of $y={\text{sin}}^{-1}(x)$, we apply differentiation rules specific to inverse trigonometric functions. The derivative formula for the inverse sine is expressed as $\frac{dy}{dx}=\frac{1}{\text{[}1-x^2{\text{]}}^{1/2}}$, where $x$ is within the domain of the inverse sine function, that is between -1 and 1.

This formula is derived using implicit differentiation and trigonometric identities. Understanding this concept allows students to analyze functions involving inverse sine in terms of their instantaneous rates of change, which is critical in many areas of science and engineering.

Chain Rule

The chain rule is among the most critical operations in calculus, specifically when it comes to differentiation. The chain rule provides a method for computing the derivative of the composition of two or more functions.

Consider two functions $f(x)$ and $g(x)$, where one function is nested within another, such as $f(g(x))$. The chain rule states that the derivative of this composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function itself. Mathematically represented as $(f\circ g{)}^{\prime }(x)=f^{\prime }(g(x))\times g^{\prime }(x)$.

For the inverse trigonometric functions, the chain rule helps in managing more complex expressions where the argument of the inverse function is not just $x$, but another function of $x$. It also ties in with implicit differentiation, which is a technique that applies the chain rule in scenarios where you have equations that can't be easily solved for one variable.

Differentiation

Differentiation is the process of finding the derivative of a function. It is a fundamental tool in calculus that measures how a function changes as its input changes. It allows us to find the rate at which variables change with respect to one another.

The derivative is often represented as $\frac{dy}{dx}$ which can be interpreted as the rate of change of $y$ with respect to $x$. Differentiation can involve straightforward cases where direct application of formulas is sufficient, and more complex scenarios where rules such as the chain rule, product rule, and quotient rule are necessary.

By differentiation, we can find tangents to curves, optimize functions, and solve problems involving motion and growth. In our exercise, determining the slope of the tangent line at a given point is accomplished through differentiation using the derivative of the inverse sine and understanding how rates of change are represented geometrically by the steepness of the tangent line.