Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the derivative of the following function.

y=6^(-4x^(2))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=6^{-4 x^{2}}$ $\newline$ Answer: $y^{\prime}=$

View step-by-step help

View step-by-step help

Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=6^{-4 x^{2}}

\newline

Answer:

y^{\prime}=

Rewrite function: We are given the function $y=6^{-4x^{2}}$ . To find the derivative, we will use the chain rule and the exponential rule for differentiation.
Apply chain rule: First, let's rewrite the function using the natural exponential function for convenience in differentiation: $\newline$ $y = e^{\ln(6)\cdot(-4x^2)}$
Differentiate with $\ln(6)$ : Now, let's differentiate both sides of the equation with respect to $x$ using the chain rule: $\newline$ $\frac{dy}{dx} = \frac{d}{dx} \left[e^{\ln(6)\cdot(-4x^2)}\right]$
Simplify exponent differentiation: Applying the chain rule, we get: $\frac{dy}{dx} = e^{\ln(6)\cdot(-4x^2)} \cdot \frac{d}{dx} [\ln(6)\cdot(-4x^2)]$
Differentiate $-4x^2$ : Since $\ln(6)$ is a constant, we can simplify the differentiation of the exponent as follows: $\newline$ $\frac{dy}{dx} = e^{\ln(6)\cdot(-4x^2)} \cdot \ln(6) \cdot \frac{d}{dx} [-4x^2]$
Substitute back derivative: Now, differentiate $-4x^2$ with respect to $x$ : $\frac{d}{dx} [-4x^2] = -4 \times 2x = -8x$
Rewrite in terms of y: Substitute the derivative of $-4x^2$ back into the equation: $\newline$ $\frac{dy}{dx} = e^{(\ln(6)\cdot(-4x^2))} \cdot \ln(6) \cdot (-8x)$
Substitute original y: Finally, we can rewrite the derivative in terms of the original function $y$ : $\frac{dy}{dx} = y \cdot \ln(6) \cdot (-8x)$
Simplify final answer: Now, we can substitute back the original expression for $y$ to get the final derivative: $\frac{dy}{dx} = 6^{(-4x^2)} \cdot \ln(6) \cdot (-8x)$
Simplify final answer: Now, we can substitute back the original expression for $y$ to get the final derivative: $\frac{dy}{dx} = 6^{(-4x^2)} \cdot \ln(6) \cdot (-8x)$ Simplify the expression to get the final answer: $y' = -8x \cdot \ln(6) \cdot 6^{(-4x^2)}$

More problems from Find derivatives of radical functions

Question

Find the derivative of

f(x)

\newline

f(x) = \cos(x)

\newline

f'(x) =

Posted 1 month ago

Question

Find the derivative of

f(x)

\newline

f(x) = \tan(x)

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = e^x

\newline

f'\left(x\right) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \ln(x)

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \tan^{-1}{x}

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = x e^x

\newline

f'\left(x\right) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \frac{e^x}{x}

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = e^{(x + 1)}

\newline

f'\left(x\right) =

Posted 1 month ago

Question

Find the derivative of

f(x)

\newline

f(x) = \sqrt{x+3}

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

g(x)

\newline

g(x) = \ln(2x)

\newline

g^{\prime}(x) =

Posted 3 months ago