Tutorial Exercise
Use logarithmic differentiation to find the derivative of the function.

Step 1
Using properties of logarithms, we can rewrite the equation as , which is equivalent to
.
SUBMIT
SKIP (YOU CANNOT COME BACK)

Ask by Davies Pritchard. in the United States

Mar 20,2025

Question

Tutorial Exercise
Use logarithmic differentiation to find the derivative of the function.

Step 1
Using properties of logarithms, we can rewrite the equation as , which is equivalent to
.
SUBMIT
SKIP (YOU CANNOT COME BACK)

Ask by Davies Pritchard. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

Let 
$$
y = x^{7\cos(x)}.
$$

**Step 1.** Take the natural logarithm of both sides:
$$
\ln(y) = \ln\left(x^{7\cos(x)}\right).
$$
Using the logarithm property $$\ln(a^b)=b\ln(a)$$, we rewrite the right-hand side:
$$
\ln(y) = 7\cos(x) \cdot \ln(x).
$$

**Step 2.** Differentiate both sides with respect to $$x$$.

Differentiate the left-hand side using the chain rule:
$$
\frac{d}{dx}[\ln(y)] = \frac{1}{y} \frac{dy}{dx}.
$$

Differentiate the right-hand side, which involves the product $$7\cos(x)\ln(x)$$. The constant $$7$$ can be factored out:
$$
\frac{d}{dx}[7\cos(x)\ln(x)] = 7\,\frac{d}{dx}[\cos(x)\ln(x)].
$$
Now, apply the product rule for $$\cos(x)\ln(x)$$, where if $$u(x)=\cos(x)$$ and $$v(x)=\ln(x)$$, then:
$$
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x).
$$

Compute:
$$
u'(x)= -\sin(x) \quad \text{and} \quad v'(x)= \frac{1}{x}.
$$
So,
$$
\frac{d}{dx}[\cos(x)\ln(x)] = -\sin(x)\ln(x) + \cos(x)\frac{1}{x}.
$$

Thus, the derivative of the right-hand side becomes:
$$
7\left(-\sin(x)\ln(x) + \frac{\cos(x)}{x}\right).
$$

**Step 3.** Equate the derivatives:
$$
\frac{1}{y}\frac{dy}{dx} = 7\left(\frac{\cos(x)}{x} - \sin(x)\ln(x)\right).
$$

Multiply both sides by $$y$$ to solve for $$\frac{dy}{dx}$$:
$$
\frac{dy}{dx} = y \cdot 7\left(\frac{\cos(x)}{x} - \sin(x)\ln(x)\right).
$$

**Step 4.** Substitute back for $$y$$:
$$
\frac{dy}{dx} = x^{7\cos(x)} \cdot 7\left(\frac{\cos(x)}{x} - \sin(x)\ln(x)\right).
$$

The final answer is:
$$
\boxed{\frac{dy}{dx} = 7x^{7\cos(x)}\left(\frac{\cos(x)}{x} - \sin(x)\ln(x)\right)}.
$$
The derivative of $$ y = x^{7\cos(x)} $$ is: $$ \frac{dy}{dx} = 7x^{7\cos(x)}\left(\frac{\cos(x)}{x} - \sin(x)\ln(x)\right). $$

Tutorial Exercise
Use logarithmic differentiation to find the derivative of the function.

Step 1
Using properties of logarithms, we can rewrite the equation as , which is equivalent to
.
SUBMIT
SKIP (YOU CANNOT COME BACK)

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Tutorial Exercise Use logarithmic differentiation to find the derivative of the function. Step 1 Using properties of logarithms, we can rewrite the equation as , which is equivalent to . SUBMIT SKIP (YOU CANNOT COME BACK)

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Tutorial Exercise
Use logarithmic differentiation to find the derivative of the function.

Step 1
Using properties of logarithms, we can rewrite the equation as , which is equivalent to
.
SUBMIT
SKIP (YOU CANNOT COME BACK)