If `y = 25^(log_5 sin x ) +16^(log_4cos x ) " then " (dy)/(dx) `= ______________

Options

A. ` 1`
B. ` 0 `
C. ` 9 `
D. ` cos x - sin x`

Answer

To find the derivative \(\frac{dy}{dx}\) for the function given by

\[ y = 25^{\log_5(\sin x)} + 16^{\log_4(\cos x)}, \]

we will simplify the expression step by step.

### Step 1: Rewrite the bases in terms of powers
We can express \(25\) and \(16\) as powers of \(5\) and \(4\) respectively:

\[ 25 = 5^2 \quad \text{and} \quad 16 = 4^2. \]

Thus, we can rewrite \(y\) as:

\[ y = (5^2)^{\log_5(\sin x)} + (4^2)^{\log_4(\cos x)}. \]

### Step 2: Apply the power of a power rule
Using the power of a power property \((a^m)^n = a^{mn}\), we can simplify further:

\[ y = 5^{2 \log_5(\sin x)} + 4^{2 \log_4(\cos x)}. \]

### Step 3: Use the property of logarithms
Using the property of logarithms that states \(a^{\log_a(b)} = b\), we can simplify each term:

\[ y = \sin^2 x + \cos^2 x. \]

### Step 4: Use the Pythagorean identity
We know from trigonometric identities that:

\[ \sin^2 x + \cos^2 x = 1. \]

Thus, we have:

\[ y = 1. \]

### Step 5: Differentiate \(y\) with respect to \(x\)
Now, we differentiate \(y\) with respect to \(x\):

\[ \frac{dy}{dx} = \frac{d}{dx}(1) = 0. \]

### Final Answer
Hence,

\[ \frac{dy}{dx} = 0. \]

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