Rational Exponents and Derivatives

The video tutorial explains how to find derivatives of various functions, starting with constant functions, then moving to square root and cube root functions using rational exponents. It covers the application of the power rule and rewriting derivatives with positive exponents. The tutorial concludes with finding the derivative of a complex function involving multiple terms and exponents, emphasizing the importance of expressing derivatives in simplified forms.

What is the derivative of a constant function with respect to x?

• It is equal to the constant itself.

• It is equal to one.

• It is undefined.

• It is equal to zero.

How can the square root of x be expressed using rational exponents?

• x to the power of 2

• x to the power of 1/2

• x to the power of 3/2

• x to the power of 1/3

What is the derivative of x to the power of 1/2 using the power rule?

• 2 times x to the power of -1/2

• 1/2 times x to the power of -1/2

• 1/2 times x to the power of 1/2

• 2 times x to the power of 1/2

How can x to the power of -1/2 be rewritten using positive exponents?

• x to the power of -1/2

• 1 divided by x to the power of 1/2

• x to the power of 1/2

• 1 divided by x to the power of -1/2

How is the cube root of x squared expressed as a rational exponent?

• x to the power of 1/3

• x to the power of 3/2

• x to the power of 2/3

• x to the power of 2

What is the derivative of x to the power of 2/3 using the power rule?

• 2/3 times x to the power of 1/3

• 2/3 times x to the power of -1/3

• 3/2 times x to the power of -1/3

• 3/2 times x to the power of 1/3

How can x to the power of -1/3 be rewritten using positive exponents?

• x to the power of 1/3

• 1 divided by x to the power of 1/3

• 1 divided by x to the power of -1/3

• x to the power of -1/3

What is the derivative of a constant term like 2e^4?

• It is equal to zero.

• It is equal to one.

• It is equal to 2e^4.

• It is undefined.

How is the term 5 divided by x squared rewritten using a negative exponent?

• 5 times x to the power of -2

• 5 times x to the power of 2

• 5 divided by x to the power of -2

• 5 divided by x to the power of 2

What is the derivative of x to the power of 1/4 using the power rule?

• 1/4 times x to the power of -3/4

• 1/4 times x to the power of 3/4

• 4 times x to the power of -3/4

• 4 times x to the power of 3/4