91Ó°ÊÓ

In Exercises 57-64 use the calculation thought experiment to say whether the expression is written as a sum, difference, scalar multiple, product, or quotient. Then use the appropriate rules to find its derivative. HINT [See Quick Examples on page 814 and Example 7.] \(y=x^{4}-\left(x^{2}+120\right)(4 x-1)\)

Find the slope of the tangent to the graph of the given function at the indicated point. HINT [Recall that the slope of the tangent to the graph of \(f\) at \(x=a\) is \(\left.f^{\prime}(a) .\right]\) $$ g(t)=\frac{1}{t^{5}} ;(1,1) $$

Find the slope of the tangent to the graph of the given function at the indicated point. HINT [Recall that the slope of the tangent to the graph of \(f\) at \(x=a\) is \(\left.f^{\prime}(a) .\right]\) $$ s(t)=\frac{1}{t^{3}} ;\left(-2,-\frac{1}{8}\right) $$

Compute the indicated derivative using the chain rule. HINT [See Quick Examples on page 828.] \(x=1-t / 2, y=4 t-1 ; \frac{d y}{d x}\)

Use the calculation thought experiment to say whether the expression is written as a sum, difference, scalar multiple, product, or quotient. Then use the appropriate rules to find its derivative. HINT [See Quick Examples on page 814 and Example 7.] \(y=x^{4}-\frac{x^{2}+120}{4 x-1}\)

Use the calculation thought experiment to say whether the expression is written as a sum, difference, scalar multiple, product, or quotient. Then use the appropriate rules to find its derivative. HINT [See Quick Examples on page 814 and Example 7.] \(y=x+1+2\left(\frac{x}{x+1}\right)\)

In Exercises 59-64, find the equation of the tangent line to the graph of the given function at the point with the indicated \(x\) -coordinate. In each case, sketch the curve together with the appropriate tangent line. $$ f(x)=x^{3} ; x=-1 $$

Compute the indicated derivative using the chain rule. HINT [See Quick Examples on page 828.] \(y=3 x^{2}-2 x ;\left.\frac{d x}{d y}\right|_{x=1}\)

Use the calculation thought experiment to say whether the expression is written as a sum, difference, scalar multiple, product, or quotient. Then use the appropriate rules to find its derivative. HINT [See Quick Examples on page 814 and Example 7.] \(y=(x+2)-4\left(x^{2}-x\right)\left(x+\frac{1}{x}\right)\) (Do not simplify the answer.)

Find the equation of the tangent line to the graph of the given function at the point with the indicated \(x\) -coordinate. In each case, sketch the curve together with the appropriate tangent line. $$ f(x)=x^{2} ; x=0 $$