Find dt/ds s=4t^3-3t^4

$s = 4 t^{3} - 3 t^{4}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d s} (s) = \frac{d}{d s} (4 t^{3} - 3 t^{4})$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d s} [s^{n}]$ is $n s^{n - 1}$ where $n = 1$ .

$1$

Step 3

Differentiate the right side of the equation.

By the Sum Rule, the derivative of $4 t^{3} - 3 t^{4}$ with respect to $s$ is $\frac{d}{d s} [4 t^{3}] + \frac{d}{d s} [- 3 t^{4}]$ .

$\frac{d}{d s} [4 t^{3}] + \frac{d}{d s} [- 3 t^{4}]$

Evaluate $\frac{d}{d s} [4 t^{3}]$ .

Since $4$ is constant with respect to $s$ , the derivative of $4 t^{3}$ with respect to $s$ is $4 \frac{d}{d s} [t^{3}]$ .

$4 \frac{d}{d s} [t^{3}] + \frac{d}{d s} [- 3 t^{4}]$

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = s^{3}$ and $g (s) = t$ .

To apply the Chain Rule, set $u_{1}$ as $t$ .

$4 (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d s} [t]) + \frac{d}{d s} [- 3 t^{4}]$

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$4 (3 {u_{1}}^{2} \frac{d}{d s} [t]) + \frac{d}{d s} [- 3 t^{4}]$

Replace all occurrences of $u_{1}$ with $t$ .

$4 (3 t^{2} \frac{d}{d s} [t]) + \frac{d}{d s} [- 3 t^{4}]$

Rewrite $\frac{d}{d s} [t]$ as $t'$ .

$4 (3 t^{2} t') + \frac{d}{d s} [- 3 t^{4}]$

Multiply $3$ by $4$ .

$12 t^{2} t' + \frac{d}{d s} [- 3 t^{4}]$

Evaluate $\frac{d}{d s} [- 3 t^{4}]$ .

Since $- 3$ is constant with respect to $s$ , the derivative of $- 3 t^{4}$ with respect to $s$ is $- 3 \frac{d}{d s} [t^{4}]$ .

$12 t^{2} t' - 3 \frac{d}{d s} [t^{4}]$

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = s^{4}$ and $g (s) = t$ .

To apply the Chain Rule, set $u_{2}$ as $t$ .

$12 t^{2} t' - 3 (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d s} [t])$

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$12 t^{2} t' - 3 (4 {u_{2}}^{3} \frac{d}{d s} [t])$

Replace all occurrences of $u_{2}$ with $t$ .

$12 t^{2} t' - 3 (4 t^{3} \frac{d}{d s} [t])$

Rewrite $\frac{d}{d s} [t]$ as $t'$ .

$12 t^{2} t' - 3 (4 t^{3} t')$

Multiply $4$ by $- 3$ .

$12 t^{2} t' - 12 t^{3} t'$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = 12 t^{2} t' - 12 t^{3} t'$

Step 5

Solve for $t'$ .

Rewrite the equation as $12 t^{2} t' - 12 t^{3} t' = 1$ .

$12 t^{2} t' - 12 t^{3} t' = 1$

Factor $12 t^{2} t'$ out of $12 t^{2} t' - 12 t^{3} t'$ .

Factor $12 t^{2} t'$ out of $12 t^{2} t'$ .

$12 t^{2} t' \cdot 1 - 12 t^{3} t' = 1$

Factor $12 t^{2} t'$ out of $- 12 t^{3} t'$ .

$12 t^{2} t' \cdot 1 + 12 t^{2} t' (- t) = 1$

Factor $12 t^{2} t'$ out of $12 t^{2} t' \cdot 1 + 12 t^{2} t' (- t)$ .

$12 t^{2} t' (1 - t) = 1$

Divide each term in $12 t^{2} t' (1 - t) = 1$ by $12 t^{2} (1 - t)$ and simplify.

Divide each term in $12 t^{2} t' (1 - t) = 1$ by $12 t^{2} (1 - t)$ .

$\frac{12 t^{2} t' (1 - t)}{12 t^{2} (1 - t)} = \frac{1}{12 t^{2} (1 - t)}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 t^{2} t' (1 - t)}{12 t^{2} (1 - t)} = \frac{1}{12 t^{2} (1 - t)}$

Rewrite the expression.

$\frac{t^{2} t' (1 - t)}{t^{2} (1 - t)} = \frac{1}{12 t^{2} (1 - t)}$

Cancel the common factor of $t^{2}$ .

Cancel the common factor.

$\frac{t^{2} t' (1 - t)}{t^{2} (1 - t)} = \frac{1}{12 t^{2} (1 - t)}$

Rewrite the expression.

$\frac{t' (1 - t)}{1 - t} = \frac{1}{12 t^{2} (1 - t)}$

Cancel the common factor of $1 - t$ .

Cancel the common factor.

$\frac{t' (1 - t)}{1 - t} = \frac{1}{12 t^{2} (1 - t)}$

Divide $t'$ by $1$ .

$t' = \frac{1}{12 t^{2} (1 - t)}$

Step 6

Replace $t'$ with $\frac{d t}{d s}$ .

$\frac{d t}{d s} = \frac{1}{12 t^{2} (1 - t)}$

Find dt/ds s=4t^3-3t^4

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