Find the derivative of the given function.h(x)=sin(log4(x2))h\left(x\right)=\sin\left(\log_4\left(x^2\right)\right)h(x)=sin(log4(x2))

2 cos ⁡ ( log ⁡ 4 x 2 ) x ln ⁡ 4 \frac{2\cos\left(\log_4x^2\right)}{x\ln4} x l n 4 2 c o s ( l o g 4 x 2 )

Find the derivative of the given function. h ( x ) = sin ( log 4 ( x 2 ) ) h\left(x\right)=\sin\left(\log_4\left(x^2\right)\right) h ( x ) = sin ( lo g 4 ( x 2 ) )

this problem, we wanna find the derivative of the function h of x is equal to the sine of log base 4 of x squared. Now here, it's pretty clear to see that we're going to have to apply the chain rule because I have a function inside of a function inside of another function. So let's go ahead and get into things here in finding this derivative, h prime of x. Now starting with our outermost function, remember that's what we like to do using the chain rule, the derivative of sine is cosine. So this is going to be cosine of everything in that argument. So log base 4 of x squared. Now from here, working our way inside using the chain rule, we are multiplying by the derivative of that next function. So here, log base 4 of x squared. Now the remember that the derivative of log base 4 of something is a one over that something. So in this case, x squared times the natural log of that base, which here is 4. Then finally, working our way to that innermost function, which is x squared, what is the derivative of x squared? We're multiplying here by 2x. And we can do a little bit of cleanup here because this x cancels with one of my x's on the bottom. And I can go ahead and rewrite this with my 2 out front. So this is 2 times the cosine of log base 4 of x squared over x times the natural log of 4. And this is my final derivative here, having applied the change rule multiple times. Let us know if you have any questions, and let's keep going.

Find the derivative of the given function. h ( x ) = sin ⁡ ( log ⁡ 4 ( x 2 ) ) h\left(x\right)=\sin\left(\log_4\left(x^2\right)\right) h ( x ) = sin ( lo g 4 ( x 2 ) )

2 cos ⁡ ( log ⁡ 4 x 2 ) x ln ⁡ 4 \frac{2\cos\left(\log_4x^2\right)}{x\ln4} x l n 4 2 c o s ( l o g 4 x 2 )

cos ⁡ ( log ⁡ 4 x 2 ) x 2 ln ⁡ 4 \frac{\cos\left(\log_4x^2\right)}{x^2\ln4} x 2 l n 4 c o s ( l o g 4 x 2 )

cos ⁡ ( log ⁡ 4 x 2 ) \cos\left(\log_4x^2\right) cos ( lo g 4 x 2 )

2 cos ⁡ ( log ⁡ 4 x 2 ) ln ⁡ 4 \frac{2\cos\left(\log_4x^2\right)}{\ln4} l n 4 2 c o s ( l o g 4 x 2 )

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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Multiple Choice

Find the derivative of the given function.
$h\left(x\right)=\sin\left(\log_4\left(x^2\right)\right)$

$\frac{\cos\left(\log_4x^2\right)}{x^2\ln4}$

$\frac{2\cos\left(\log_4x^2\right)}{x\ln4}$

$\cos\left(\log_4x^2\right)$

$\frac{2\cos\left(\log_4x^2\right)}{\ln4}$

Verified step by step guidance

Identify the function h(x) = sin(log_4(x^2)). This is a composite function involving the sine function and the logarithm function.

Apply the chain rule for differentiation. The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) * g'(x).

Differentiate the outer function, which is sin(u), where u = log_4(x^2). The derivative of sin(u) with respect to u is cos(u).

Differentiate the inner function u = log_4(x^2). Use the change of base formula for logarithms: log_4(x^2) = log(x^2) / log(4). The derivative of log(x^2) with respect to x is 2/x, using the chain rule.

Combine the derivatives using the chain rule: h'(x) = cos(log_4(x^2)) * (2/x) / ln(4). This gives the derivative of the original function.

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