Find the derivative of the function using the definition of derivative. State the domain of the

step 1 F of x is equal to x to the power 3 over 2. So first thing about the domain. So the domain of th

Find the derivative of the function using the definition of...

Key Concepts

Limit Definition of Derivative

The limit definition of the derivative is a fundamental concept in calculus where the derivative of a function at a point is defined as the limit of the difference quotient as the increment approaches zero. It is expressed as f'(x) = lim(h?0)[f(x+h) - f(x)]/h, and this definition provides the foundation for understanding rates of change and slopes of tangent lines.

Fractional Exponents

Fractional exponents represent a combination of powers and roots. For example, an exponent of 3/2 means taking the square root of the base and then raising it to the third power. This interpretation impacts both the computation of derivatives and the determination of the function's domain, since roots typically impose restrictions such as non-negativity of the base under real number definitions.

Domain Analysis in Functions and Derivatives

Domain analysis involves determining the set of input values for which a function is defined, as well as identifying where its derivative exists. For functions with fractional exponents, the domain is often restricted (for instance, to non-negative numbers when dealing with square roots). This analysis is crucial when stating the domain of the function and ensuring that the derivative is computed only within the interval where the original function is defined.

Transcript

00:01 F of x is equal to x to the power 3 over 2.

00:06 So first thing about the domain.

00:08 So the domain of this function is all real numbers.

00:13 So the domain is from negative infinity to positive infinity.

00:17 Now the next thing we need to do is we need to find the derivative.

00:21 So f of x by definition is limit h tends to 0, f of x plus h minus f of x over so that would be equal to limit h tends to 0, x plus h to the power 3 over 2 minus x to the power 3 over 2 over h.

00:46 So what we're going to do here is we're going to rewrite this slightly differently.

00:51 So we're going to say this is limit h tends to 0 x plus h to the power 1 half, raised to the power 3 and minus x power 1 half raised to the power 3 over h so this now is of the form a squared minus b a cube minus b cubed so going back to our basic factoring a cubed minus b cubed can be factored as a minus b times a squared plus a plus b squared.

01:35 So what that means is if i say a is equal to x plus h to the power one half, and i say b is equal to x to the power one half, then i can factor this.

01:51 X plus h to the power one half raised to the power three minus x power one half raised to the power three can be factored as we said a minus b so that would be x plus h power one half minus x power one half and then that would be times a squared so a squared would be x plus h and then plus ab so that would be plus x plus h to the power one half times x to the power one half and and then plus b squared.

02:36 So that would be plus x.

02:39 So this means that now we can replace this year with that expression.

02:46 So we have limit h tends to 0.

02:50 And this would be x plus h power 1 half, minus x power 1 half, and then times.

03:02 So x plus h.

03:05 Plus x plus h power one half squared times x power one half half and then plus x or h so we can further simplify this just a little bit so we'll say limit h tends to zero and then x plus h power one half minus x power one half and then times this becomes 2x plus h and then times plus x plus h power one half times x power one half over h so far so good now we're going to use something called the commutative property of multiplication which simply says that the two factors are being multiplied, then you can switch their places.

04:17 So i'm going to write this as 2x plus h plus x plus h power one half times x power one half and then times x plus h power one half.

04:34 So all i've done here is interchange the positions of those two factors...

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