If y = (sin x)^tan x, then dy/dx is equal to (A) (sin x)^tan x (1 + sec² x log sin x) (B) tan x (sin

Q: Wait, why can't I just use the power rule and write $\tan x (\sin x)^{\tan x - 1}$?

That is the classic trap! The power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ assumes the exponent $n$ is a constant. In your function, the exponent $\tan x$ is a variable that changes as $x$ changes. You cannot treat it as a constant. You must use logarithmic differentiation to handle variable-on-variable expressions.

Q: Why do we use $\ln$ (natural log) instead of $\log_{10}$?

In calculus, $\ln$ is the standard because its derivative is simple: $\frac{d}{dx}(\ln x) = \frac{1}{x}$. If you used $\log_{10}$, you would introduce an extra constant factor of $\frac{1}{\ln 10}$ due to the change-of-base formula, which would make your derivative unnecessarily messy. Stick with $\ln$!

Q: Do I really need to substitute $y$ back at the end?

Absolutely! The question asks for $\frac{dy}{dx}$ in terms of $x$. If you leave $y$ in your final answer, it is incomplete. Always substitute the original function $y = (\sin x)^{\tan x}$ back into your final expression to provide the answer in the requested form.

Question

If $y = (\sin x)^{	an x}$, then $\frac{dy}{dx}$ is equal to

Options:

(A) $(\sin x)^{	an x} (1 + \sec^2 x \log \sin x)$

(B) $	an x (\sin x)^{	an x-1} \cdot \cos x$

(C) $(\sin x)^{	an x} \sec^2 x \log \sin x$

(D) $	an x (\sin x)^{	an x-1}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The function $y = (\sin x)^{	an x}$ is a classic example of a 'variable-on-variable' function. The standard power rule, $\frac{d}{dx}(x^n) = nx^{n-1}$, is inapplicable here because the exponent $	an x$ is not a constant.

To differentiate this, we must employ the **Logarithmic Bridge**. This technique allows us to convert the variable exponent into a coefficient, making the expression manageable for standard differentiation rules.

### The Logarithmic Transformation

We begin by taking the natural logarithm of both sides:
$$\ln y = \ln((\sin x)^{	an x})$$

Using the logarithmic power property $\ln(a^b) = b \cdot \ln a$, we bring the exponent down:
$$\ln y = 	an x \cdot \ln(\sin x)$$

The expression is now a product of two functions, which is significantly easier to differentiate than an exponential form.

### The Calculus Dance

We differentiate both sides with respect to $x$. For the left side, we apply the chain rule:
$$\frac{d}{dx}(\ln y) = \frac{1}{y} \cdot \frac{dy}{dx}$$

For the right side, we apply the **Product Rule**, where $u = 	an x$ and $v = \ln(\sin x)$:
$$\frac{d}{dx}(	an x \cdot \ln(\sin x)) = 	an x \cdot \frac{d}{dx}(\ln(\sin x)) + \ln(\sin x) \cdot \frac{d}{dx}(	an x)$$

Calculating the individual derivatives, we find:
$$\frac{d}{dx}(\ln(\sin x)) = \frac{1}{\sin x} \cdot \cos x = \cot x$$
$$\frac{d}{dx}(	an x) = \sec^2 x$$

### The Final Synthesis

Substituting these back into our product rule equation, we obtain:
$$\frac{1}{y} \frac{dy}{dx} = 	an x \cdot (\cot x) + \ln(\sin x) \cdot (\sec^2 x)$$

Since $	an x \cdot \cot x = 1$, the expression simplifies to:
$$\frac{1}{y} \frac{dy}{dx} = 1 + \sec^2 x \cdot \ln(\sin x)$$

To isolate $\frac{dy}{dx}$, we multiply both sides by $y$:
$$\frac{dy}{dx} = y \cdot [1 + \sec^2 x \cdot \ln(\sin x)]$$

Finally, substituting the original definition of $y = (\sin x)^{	an x}$, we arrive at the **final derivative**:
$$\frac{dy}{dx} = (\sin x)^{	an x} \cdot (1 + \sec^2 x \cdot \ln(\sin x))$$

Animated Solution for Mathematics - Differentiation: If $y = (sin x)^{t a n x}$ , then $\frac{d y}{d x}$ is equal to

Select Answer:

Visualized Solution

The Function and Its Derivative

Identifying the Form

Logarithmic Differentiation

Simplifying the Exponent

Differentiating Both Sides

Differentiating the Left Side

Setting up the Product Rule

Applying the Product Rule

Differentiating the Second Term

Differentiating the First Term

Substituting Derivatives Back

Simplifying the Expression

Isolating $\frac{d y}{d x}$

The Final Answer

The Sigma Insight: Techniques of Differentiation

Analyzing the Setup

The Logarithmic Transformation

The Calculus Dance

The Final Synthesis

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Animated Solution for Mathematics - Differentiation: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If y=(sinx)tanx, then dxdy​ is equal to