The following diagram shows the derivatives of exponential functions. Do not confuse it with the function g(x) x 2, in which the variable is the base.
And now the derivative of this term is 1 over 1 plus x squared times the derivative of 1 plus x² which is 2x. On the basis of the logarithm function, Base 10 and 10x b. 1 over 0.5x times the derivative of 0.5x plus 1 and that’s 0.5 plus. The log functions of 0 to the base 10 is expressed as. The function f(x) 2 x is called an exponential function because the variable x is the variable. H(x) is going to be, and according to our general logarithmic rule, its 1 over the inside part. So to find the second derivative of sin^2x, we just need to differentiate 3sin 2(x)cos(x). Related Pages Exponential Functions Derivative Rules Natural Logarithm Calculus Lessons. To calculate the second derivative of a function, you just differentiate the first derivative.įrom above, we found that the first derivative of sin^3x = 3sin 2(x)cos(x).
► Derivative of sin x cubed = 3sin 2(x)cos(x) ► Derivative of sin cubed x = 3sin 2(x)cos(x) ► Derivative of (sinx)^3 = 3sin 2(x)cos(x) Lets do a little work with the definition again: d dx ax lim x0 ax+x ax x lim x0 axax ax x lim x0ax ax 1 x ax lim x. ► Derivative of sin 3 x = 3sin 2(x)cos(x) As with the sine function, we dont know anything about derivatives that allows us to compute the derivatives of the exponential and logarithmic functions without going back to basics. Just be aware that not all of the forms below are mathematically correct. Using the chain rule, the derivative of sin^3(x) is 3sin 2(x)cos(x)įinally, just a note on syntax and notation: sin^3(x) is sometimes written in the forms below (with the derivative as per the calculations above).
How to find the derivative of sin^3(x) using the Chain Rule: F'(x)į(g(x)) = (sin(x)) 3 ⇒ f'(g(x)) = 3sin 2(x)
Now we can just plug f(x) and g(x) into the chain rule. Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x) We can find the derivative of sin^3x (F'(x)) by making use of the chain rule.įor two differentiable functions f(x) and g(x) Use the Exponential Function (on both sides): 3(log3(x))35. Let’s define this composite function as F(x): The Exponent takes 2 and 3 and gives 8 (2, used 3 times in a multiplication, makes 8). So if the function f(x) = x 3 and the function g(x) = sin(x), then the function (sin(x)) 3 can be written as a composite function. Let’s call the function of the base g(x), which means: Now the function is in the form of x 3, except it does not have x as the base, instead it has another function of x (sin(x)) as the base. Using the chain rule to find the derivative of sin^3(x)Īlthough the expression sin 3x contains no parenthesis, we can still view it as a composite function (a function of a function).
This means the chain rule will allow us to perform the differentiation of the expression sin^3(x). We know how to differentiate x 3 (the answer is 3x 2).We know how to differentiate sin(x) (the answer is cos(x)).The chain rule is useful for finding the derivative of a function which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own. Here is our post dealing with how to differentiate sin(3x). They have simple derivatives, so they are often used in the solution of. Note that in this post we will be looking at differentiating sin 3(x) which is not the same as differentiating sin(3x). ( x) x e y 1 d d x ( e y) 1 e y d y d x d y d x 1 e y d y d x 1 x. Or, the logarithm of 81 to the base 3 is 4, because 3 raised to the power of 4. How to calculate the derivative of sin^3x Graphically this means that they have the same graph except that one is “flipped” or “reflected” through the line \(y=x\) as shown in Figure 4.5.The derivative of sin^3(x) is 3sin^2(x)cos(x) \frac\) which as you probably know is often abbreviated \(\ln x\) and called the “natural logarithm” function.Ĭonsider the relationship between the two functions, namely, that they are inverses, that one “undoes” the other. Implicit and Logarithmic Differentiation.Derivatives of Exponential & Logarithmic Functions.Derivative Rules for Trigonometric Functions.Limits at Infinity, Infinite Limits and Asymptotes.Symmetry, Transformations and Compositions.Open Educational Resources (OER) Support: Corrections and Suggestions.