We can see that our answers are correct when we graph the curve which is a parabola and observe the slopes of the tangents. This is what makes calculus so powerful.

## The Definition of the Derivative

We can find the slope anywhere on the curve i. Derivative of square root of sine x by first principles. Can we find the derivative of all functions? How to differentiate? Rate of change by Rismiya [Solved! More info: Calculus videos. Sign up for the free IntMath Newsletter.

## 3. The Derivative from First Principles

Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents! Limits and Differentiation 2. The Slope of a Tangent to a Curve Numerical 3.

The Derivative from First Principles 4. Derivative as an Instantaneous Rate of Change 5. Derivatives of Polynomials 5a. Derivative interactive graphs - polynomials 6.

Derivatives of Products and Quotients 7. Differentiating Powers of a Function 7a.

Differentiation interactive applet - products and quotients 8. Differentiation of Implicit Functions 9. Higher Derivatives Partial Derivatives Feedback? The Derivative from First Principles.

## The First Derivative Rule

Previously, derivatives of algebraic functions have proven to be algebraic functions and derivatives of trigonometric functions have been shown to be trigonometric functions. Here, for the first time, we see that the derivative of a function need not be of the same type as the original function.

The derivatives of the remaining inverse trigonometric functions may also be found by using the inverse function theorem. These formulas are provided in the following theorem. The Derivative of an Inverse Function We begin by considering a function and its inverse.

We summarize this result in the following theorem. Hint Use the preceding example as a guide.

### Derivatives of Inverse Trigonometric Functions

Hint Use the chain rule. Derivatives of Inverse Trigonometric Functions We now turn our attention to finding derivatives of inverse trigonometric functions. Key Concepts The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative. We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.