Resolved: "Find derivatives of the functions defined as follows:..." - Plainmath

Dottie Parra

Answered question

2021-10-08

Find derivatives of the functions defined as follows:

y = 3 \cdot 4^{x^{2} + 2}

Answer & Explanation

Roosevelt Houghton

Skilled2021-10-09Added 106 answers

Step 1
To find the derivative of:

y = 3 \cdot 4^{x^{2} + 2}

Solution:

y = 3 \cdot 4^{x^{2} + 2}

On differentiating both sides w.r.t x we get:

\frac{d}{d x} (y) = \frac{d}{d x} (3 \cdot 4^{x^{2} + 2})

\Rightarrow \frac{d y}{d x} = 3 \cdot (4^{x^{2} + 2}) \cdot \ln (4) \cdot \frac{d}{d x} (x^{2} + 2)

(using,

\frac{d}{d x} (f (g (x))) = f^{'} (g (x)) \cdot g^{'} (x)

)

\Rightarrow \frac{d y}{d x} = 3 \cdot (4^{x^{2} + 2}) \cdot \ln (4) \cdot (2 x)

(using,

\frac{d}{d x} (a^{x}) = a^{x} \ln a, \frac{d}{d x} (x^{n}) = n x^{n - 1}

)

\Rightarrow \frac{d y}{d x} = 6 x (4^{x^{2} + 2} \cdot \ln (4))

Step 2
Result:

\frac{d y}{d x} = 6 x (4^{x^{2} + 2} \cdot \ln (4))

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