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=3^(-9x^(4)+3x^(3))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=3^{-9 x^{4}+3 x^{3}}$ $\newline$ 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=3^{-9 x^{4}+3 x^{3}}

\newline

Answer:

y^{\prime}=

Identify Function Components: Identify the function and its components. $\newline$ We have $y = 3^{(-9x^{4} + 3x^{3})}$ . This is an exponential function with base $3$ and exponent $-9x^4 + 3x^3$ .
Apply Chain Rule: Apply the chain rule for derivatives. $\newline$ The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $a^u$ where $a$ is a constant and $u$ is a function of $x$ , and the inner function is $u(x) = -9x^4 + 3x^3$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $a^u$ with respect to $u$ is $a^u \cdot \ln(a)$ , where $\ln(a)$ is the natural logarithm of $a$ . So, the derivative of $3^u$ with respect to $u$ is $3^u \cdot \ln(3)$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The inner function is $u(x) = -9x^4 + 3x^3$ . Using the power rule, the derivative of $-9x^4$ is $-36x^3$ , and the derivative of $3x^3$ is $9x^2$ . So, the derivative of $u(x)$ with respect to $x$ is $u'(x) = -36x^3 + 9x^2$ .
Apply Chain Rule Multiplication: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ Using the chain rule, the derivative of $y$ with respect to $x$ is $y' = (3^u \cdot \ln(3)) \cdot u'(x)$ , where $u = -9x^4 + 3x^3$ and $u'(x) = -36x^3 + 9x^2$ . Substituting $u$ and $u'(x)$ into the equation, we get $y' = (3^{(-9x^4 + 3x^3)} \cdot \ln(3)) \cdot (-36x^3 + 9x^2)$ .
Simplify Derivative Expression: Simplify the expression for the derivative. $y' = 3^{(-9x^4 + 3x^3)} \cdot \ln(3) \cdot (-36x^3 + 9x^2)$ can be simplified by factoring out common terms if possible. However, in this case, the expression is already in its simplest form.

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