Differentiability

To find the derivative $%5Cfrac%7Bdy%7D%7Bdx%7D$ of the function $y%20%3D%20%5Ccos%5E2%20x$ , we'll use the chain rule of differentiation.

1. Given Function:
$y%20%3D%20%5Ccos%5E2%20x$ This can be rewritten as: $y%20%3D%20(%5Ccos%20x)%5E2$

2. Apply the Chain Rule:
The chain rule states that if $y%20%3D%20%5Bf(x)%5D%5En$ , then: $%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20n%5Bf(x)%5D%5E%7Bn-1%7D%20%5Ccdot%20f'(x)$ Here, $f(x)%20%3D%20%5Ccos%20x$ and $n%20%3D%202$ .

3. Differentiate Step-by-Step:
$%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%202(%5Ccos%20x)%5E%7B2-1%7D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D(%5Ccos%20x)%20%5C%5C%0A%3D%202%5Ccos%20x%20%5Ccdot%20(-%5Csin%20x)%20%5C%5C%0A%3D%20-2%5Csin%20x%20%5Ccos%20x$

4. Simplify Using Trigonometric Identity:
We can express the result using the double-angle identity: $%5Csin%202x%20%3D%202%5Csin%20x%20%5Ccos%20x$ Thus: $%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20-%5Csin%202x$

5. Final Answer:
The derivative of $y%20%3D%20%5Ccos%5E2%20x$ with respect to $x$ is: $%5Cboxed%7B-2%5Csin%20x%20%5Ccos%20x%7D%20%5Cquad%20%5Ctext%7Bor%20equivalently%7D%20%5Cquad%20%5Cboxed%7B-%5Csin%202x%7D$

To find the derivative $%5Cfrac%7Bdy%7D%7Bdx%7D$ of the function $y%20%3D%20%5Ccos%5E2%20x$ , we'll use the chain rule of differentiation.

1. Given Function:
$y%20%3D%20%5Ccos%5E2%20x$ This can be rewritten as: $y%20%3D%20(%5Ccos%20x)%5E2$

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Differentiability

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs