2.3 Derivatives: Power Rule, Sine And Cosineap Calculus

Now we can complete the calculation of the derivative of the sine:$$eqalign{{dover dx}sin x &= lim_{Delta xto0} {sin(x+Delta x)-sin x over Delta x}cr&=lim_{Delta xto0} sin x{cos Delta x - 1over Delta x}+cos x{sinDelta xover Delta x}cr&=sin x cdot 0 + cos x cdot 1 = cos x.cr}$$

2.3 Derivatives: Power Rule Sine And Cosineap Calculus Solver
2.3 Derivatives: Power Rule Sine And Cosineap Calculus Problems
2.3 Derivatives: Power Rule Sine And Cosineap Calculus Calculator
2.3 Derivatives: Power Rule Sine And Cosineap Calculus 2nd Edition

The derivative of a function measures the slope or steepness of thefunction; if we examine the graphs of the sine and cosine side byside, it should be that the latter appears to accurately describe theslope of the former, and indeed this is true:

That is, the derivative of the co sine, co tangent, and co secant are the ones with negative signs. The trig functions are paired when it comes to differentiation: sine and cosine, tangent and secant, cotangent and cosecant. This lesson assumes you are familiar with the Power Rule, Product Rule, Quotient Rule and Chain Rule. 1.2 Calculus Without Limits 1.3 The Velocity at an Instant 1.4 Circular Motion 1.5 A Review of Trigonometry 1.6 A Thousand Points of Light 1.7 Computing in Calculus (PDF - 1.2MB) 2: Derivatives. 2.1 The Derivative of a Function 2.2 Powers and Polynomials 2.3 The Slope and the Tangent Line 2.4 Derivative of the Sine and Cosine. Calculus Maximus Notes: 2.3 Differentiation Rules Page 1 of 7 §2.3—Differentiation Rules. dy dx is a noun. It means “the derivative of y with respect to x.”. d dx is a verb. It means “take the derivative with respect to x” of the expression that follows. The Constant Rule The derivative of a constant function is 0.

2.3 Derivatives: Power Rule, Sine And Cosineap Calculus

Notice that where the cosine is zero the sine does appear to have ahorizontal tangent line, and that the sine appears to be steepestwhere the cosine takes on its extreme values of 1 and $-1$.

Of course, now that we know the derivative of the sine, we can computederivatives of more complicated functions involving the sine.

Example 4.4.1 Compute the derivative of $ds sin(x^2)$.$${dover dx}sin(x^2) = cos(x^2)cdot 2x = 2xcos(x^2).$$

Example 4.4.2 Compute the derivative of $ds sin^2(x^3-5x)$.$$eqalign{{dover dx}sin^2(x^3-5x)&={dover dx}(sin(x^3-5x))^2cr&=2(sin(x^3-5x))^1cos(x^3-5x)(3x^2-5)cr&=2(3x^2-5)cos(x^3-5x)sin(x^3-5x).cr}$$

Exercises 4.4

Find the derivatives of the following functions.

Ex 4.4.1$ds sin^2(sqrt{x})$(answer)

Ex 4.4.2$ds sqrt{x}sin x$(answer)

Ex 4.4.3$ds {1over sin x}$(answer)

Ex 4.4.4$ds {x^2+xover sin x}$(answer)