Derivatives and Rules of Differentiation

The video tutorial explains how to find the first and second derivatives of a composite function, f(x) = cos(2x^5). It begins by identifying the inner and outer functions and uses the chain rule to find the first derivative. The second derivative is calculated using the product rule, followed by simplification and factoring to express it in a more concise form.

What is the outer function in the given composite function?

• 2x to the fifth

• Cosine

• Tangent

• Sine

What is the derivative of the inner function 2x to the fifth?

• 2x to the fourth

• x to the fifth

• 5x to the fourth

• 10x to the fourth

Which rule is applied to find the first derivative of the composite function?

• Product Rule

• Power Rule

• Quotient Rule

• Chain Rule

What is the first derivative of the function f(x) = cos(2x^5)?

• -10x^4 cos(2x^5)

• 10x^4 cos(2x^5)

• -10x^4 sin(2x^5)

• 10x^4 sin(2x^5)

Which rule is used to find the second derivative of the function?

• Sum Rule

• Chain Rule

• Product Rule

• Quotient Rule

What is the derivative of sine(2x^5) with respect to x?

• cos(2x^5) * 10x^4

• sin(2x^5) * 10x^4

• cos(2x^5) * 5x^4

• sin(2x^5) * 5x^4

What is the simplified form of the second derivative?

• -20x^3(5x^5 sin(2x^5) + 2 cos(2x^5))

• 20x^3(5x^5 cos(2x^5) + 2 sin(2x^5))

• 20x^3(5x^5 sin(2x^5) + 2 cos(2x^5))

• -20x^3(5x^5 cos(2x^5) + 2 sin(2x^5))

What is the greatest common factor factored out in the second derivative?

• 20x^3

• 40x^3

• 10x^3

• 30x^3

What is the expression for the second derivative before simplification?

• -100x^8 cos(2x^5) - 40x^3 sin(2x^5)

• 100x^8 cos(2x^5) + 40x^3 sin(2x^5)

• -100x^8 sin(2x^5) - 40x^3 cos(2x^5)

• 100x^8 sin(2x^5) + 40x^3 cos(2x^5)

What is the first derivative of the function f(x) = cos(2x^5) as mentioned in the conclusion?

• -10x^4 cos(2x^5)

• 10x^4 cos(2x^5)

• 10x^4 sin(2x^5)

• -10x^4 sin(2x^5)