f(x)=sin^(7)x
The 7 is an exponent and the x is not.
Find the derivative.
Let u=sin(x)
f(x)=u7
f'(x)=d (u7) /dx
= d (u7) /du . du/dx
= 7u6 . d(sin(x))/dx
= 7 sin6(x) cos(x)
let z = sin x
then f(z) = z^7
and dz/dx = cos x
df(z)/ dz = 7 z^6
d f(x) dx = d f(z)/dz * dz/dx
= 7 z^6 cos x = 7 sin^6 x cos x
I got the same answer but for some reason, it's not the right answer(webwork). Thanks though!
I got the same answer as Damon and MathMate and I'll give 1000:1 odds that we are right.
(It could be that they have a variation of our answer. An easy way to check if two different looking answers are both correct, pick any obscure angle, e.g. 78ยบ, and sub it into both answers. If you get the same result .....)
If it's web exercise, try the equivalent forms:
7sin5(x) sin(2x) /2
or
3.5sin5(x) sin(2x)
To find the derivative of the function f(x) = sin^7(x), we can use the chain rule. The chain rule allows us to find the derivative of a composition of functions.
The chain rule states that if we have a composite function y = f(g(x)), where f(u) and g(x) are both differentiable functions, then the derivative dy/dx is given by dy/dx = f'(g(x)) * g'(x).
Let's apply the chain rule to find the derivative of f(x) = sin^7(x).
Step 1: Identify the composite function.
In the given function, f(x) = sin^7(x), the composite function is sin^7(x), where g(x) = sin(x) and f(u) = u^7.
Step 2: Find the derivative of the inner function.
The derivative of g(x) = sin(x) is g'(x) = cos(x).
Step 3: Find the derivative of the outer function.
The derivative of f(u) = u^7 with respect to u is f'(u) = 7u^6.
Step 4: Apply the chain rule.
Using the chain rule, the derivative of f(x) = sin^7(x) is:
f'(x) = f'(g(x)) * g'(x) = 7(sin(x))^6 * cos(x).
Therefore, the derivative of f(x) = sin^7(x) is f'(x) = 7(sin(x))^6 * cos(x).