Determine (d y)/(d x) for each of the following: a. y=sin2 x b. y=2 cos3 x c. y=sin(x^3-2 x+4) d

Determine (d y)/(d x) for each of the following: a. y=sin2 x...

Key Concepts

Chain_Rule

The chain rule is a fundamental technique in differentiation used to compute the derivative of composite functions. It states that if a function is composed of an outer function and an inner function, the derivative is obtained by multiplying the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function.

Sum_and_Constant_Multiple_Rules

The sum rule allows the differentiation of a sum or difference of functions term by term. The constant multiple rule states that a constant factor can be taken outside the differentiation process, meaning the derivative of a constant times a function is the constant multiplied by the derivative of the function. Together, these rules simplify differentiation of functions expressed as sums or scaled versions of basic functions.

Derivatives_of_Trigonometric_Functions

Standard trigonometric functions such as sine and cosine have established derivative formulas: the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). These formulas are applied directly or in combination with the chain rule when the argument of the trigonometric function is itself a function of x.

Derivatives_of_Exponential_Functions

Exponential functions, most notably the natural exponential function e^x, are differentiated by applying the rule that the derivative of e^x is e^x. When the exponent is a function of x, the chain rule must be used to account for the inner function’s derivative.

Derivatives_of_Power_Functions

Power functions, represented by expressions of the form x^n, obey the power rule. This rule states that the derivative of x raised to any power n is n multiplied by x raised to the power of (n-1), making it a straightforward and essential technique in calculus.

Constant_Function_Derivative

A constant function, or any term that simplifies to a constant, has a derivative of zero. Recognizing and correctly handling constant terms is important in differentiation, as they do not change with respect to the variable.

Transcript

00:01 Since there's so many problems to do, i'm just going to jump right into this that i'm differentiating each one of these.

00:07 So starting with sine of 2x.

00:09 And i'm assuming that you have some background knowledge and you can learn now that the derivative of sine is cosine.

00:15 That's a chain rule where you leave 2x alone multiplied by the derivative of 2x, which is 2.

00:22 So moving right on to letter b or taking the derivative of 2 cosine of 3x.

00:29 So again, the two that's in front just goes along for the ride.

00:34 The derivative of cosine is negative sign.

00:37 You leave 3x alone, but then you multiply by the derivative of 3x, which is 3, and then now i'm going to consider that 2 in front.

00:47 So if we look at letter c, we're taking the derivative, and we have, what is it, sign of this complicated thing, x, q minus 2x plus 4.

01:04 And again it is the chain rule where the derivative of sine is cosine, leave that piece alone, and then you have to multiply by the derivative of x cubed, and that's your power rule, and the derivative of 2x, and the derivative of 4 is 0, and just sneak that in front.

01:31 Let's see, moving on to d, we have the derivative of 2, cosine of negative 4x.

01:37 So the derivative, again, the two goes along for the ride.

01:49 The derivative of cosine is negative sign, leave negative 4x alone, but then multiply by the derivative of negative 4x, which is negative 4 times this 2 makes it positive 8.

02:03 Yeah, let's move on to the next one where we are finding the derivative.

02:11 Letter e, we have sign of 3x minus cosine of.

02:21 4x.

02:23 So again, i'm assuming you have some background knowledge that the derivative of sine is cosine, leave 3x alone, and multiplied by the derivative of 3x, which is 3.

02:33 Now, the derivative of cosine is negative sign, so it's going to change that minus to a plus, leave 4x alone, but multiplied by the derivative of 4x, 4.

02:47 Next option, we're doing 2 to the x plus to sign of x...

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