What is the derivative of (1+y^x) ?

The derivative of (1+y^x) is (ln(1+y)+(x\frac{dy)/(dx)}{1+y})(1+y)^x

What is the first derivative of (1+y^x) ?

The first derivative of (1+y^x) is (ln(1+y)+(x\frac{dy)/(dx)}{1+y})(1+y)^x

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\frac{d}{d x} ((1 + y)^{x})

(ln (1 + y) + \frac{x \frac{d y}{d x}}{1 + y}) (1 + y)^{x}

Solution steps

\frac{d}{d x} ((1 + y)^{x})

Apply exponent rule:

a^{b} = e^{b l n (a)}

(1 + y)^{x} = e^{x l n (1 + y)}

= \frac{d}{d x} (e^{x l n (1 + y)})

Apply the chain rule:

e^{x l n (1 + y)} \frac{d}{d x} (x ln (1 + y))

= e^{x l n (1 + y)} \frac{d}{d x} (x ln (1 + y))

\frac{d}{d x} (x ln (1 + y)) = ln (1 + y) + \frac{x \frac{d y}{d x}}{1 + y}

= e^{x l n (1 + y)} (ln (1 + y) + \frac{x \frac{d y}{d x}}{1 + y})

e^{x l n (1 + y)} = (1 + y)^{x}

= (ln (1 + y) + \frac{x \frac{d y}{d x}}{1 + y}) (1 + y)^{x}

(95) (25)

6 \cdot 3.13 \cdot 1 0^{- 2}

1.965401545 \cdot 1 0^{- 3}

e^{- c o s (y)}

What is the derivative of (1+y^x) ?
The derivative of (1+y^x) is (ln(1+y)+(x\frac{dy)/(dx)}{1+y})(1+y)^x
What is the first derivative of (1+y^x) ?
The first derivative of (1+y^x) is (ln(1+y)+(x\frac{dy)/(dx)}{1+y})(1+y)^x